keyword:new

A376335

Irregular triangle read by rows: T(n,k) = A008949(n-1,k) if 0 <= k <= n - 2 otherwise A008949(n-1,2*n-4-k) if n - 2 <= k <= 2*n - 4.

+0
0

1, 1, 3, 1, 1, 4, 7, 4, 1, 1, 5, 11, 15, 11, 5, 1, 1, 6, 16, 26, 31, 26, 16, 6, 1, 1, 7, 22, 42, 57, 63, 57, 42, 22, 7, 1, 1, 8, 29, 64, 99, 120, 127, 120, 99, 64, 29, 8, 1, 1, 9, 37, 93, 163, 219, 247, 255, 247, 219, 163, 93, 37, 9, 1, 1, 10, 46, 130, 256, 382, 466, 502, 511, 502, 466, 382, 256, 130, 46, 10, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,3

LINKS

Table of n, a(n) for n=2..82.

Nsibiet E. Udo, Praise Adeyemo, Balazs Szendroi, and Stavros Argyrios Papadakis, Ideals, representations and a symmetrised Bernoulli triangle, arXiv:2409.10278 [math.AC], 2024. See p. 2.

FORMULA

Sum_{k=0..2*n-4} T(n,k) = A000337(n-1). [Udo et al.]

EXAMPLE

The triangle begins as:

1;

1, 3, 1;

1, 4, 7, 4, 1;

1, 5, 11, 15, 11, 5, 1;

1, 6, 16, 26, 31, 26, 16, 6, 1;

1, 7, 22, 42, 57, 63, 57, 42, 22, 7, 1;

...

MATHEMATICA

b[n_, k_]:=Sum[Binomial[n, j], {j, 0, k}]; T[n_, k_]:=If[0<=k<=n-2, b[n-1, k], b[n-1, 2n-4-k]]; Table[T[n, k], {n, 2, 10}, {k, 0, 2n-4}]//Flatten

CROSSREFS

Cf. A000337, A008949.

KEYWORD

nonn,easy,tabf,new

AUTHOR

Stefano Spezia, Sep 20 2024

STATUS

approved

A375977

Number of subsets of {1,2,...,n} such that no two elements differ by 2 or 5.

+0
0

1, 2, 4, 6, 9, 15, 21, 29, 45, 69, 100, 152, 236, 349, 517, 789, 1185, 1757, 2653, 4014, 5992, 8986, 13573, 20363, 30485, 45901, 69041, 103481, 155468, 233908, 351104, 527033, 792405, 1190493, 1787129, 2685209, 4035261, 6059758, 9101828, 13676670, 20544125

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..40.

Michael A. Allen, Combinations without specified separations, Communications in Combinatorics and Optimization (in press).

Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,-1,1,2,-1,0,1,-1).

FORMULA

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) + 2*a(n-7) - a(n-8) + a(n-10) - a(n-11) for n >= 11.

G.f.: (1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 - x^7 - x^10)/(1 - x - x^3 + x^5 - x^6 - 2*x^7 + x^8 - x^10 + x^11).

EXAMPLE

For n = 6, the 21 subsets are {}, {1}, {2}, {1,2}, {3}, {2,3}, {4}, {1,4}, {3,4}, {5}, {1,5}, {2,5}, {1,2,5}, {4,5}, {1,4,5}, {6}, {2,6}, {3,6}, {2,3,6}, {5,6}, {2,5,6}.

MATHEMATICA

CoefficientList[Series[(1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 - x^7 - x^10)/(1 - x - x^3 + x^5 - x^6 - 2*x^7 + x^8 - x^10 + x^11), {x, 0, 39}], x]

LinearRecurrence[{1, 0, 1, 0, -1, 1, 2, -1, 0, 1, -1}, {1, 2, 4, 6, 9, 15, 21, 29, 45, 69, 100}, 39]

CROSSREFS

See A375981 for other sequences related to restricted combinations.

Column k=18 of A376033.

KEYWORD

nonn,easy,new

AUTHOR

Michael A. Allen, Sep 20 2024

STATUS

approved

A375978

Number of subsets of {1,2,...,n} such that no two elements differ by 3 or 5.

+0
0

1, 2, 4, 8, 12, 18, 24, 34, 47, 73, 111, 177, 267, 409, 600, 900, 1324, 2004, 2996, 4564, 6848, 10377, 15513, 23385, 34953, 52685, 78969, 119138, 178840, 269604, 404656, 609310, 914548, 1376530, 2067231, 3111457, 4674751, 7034897, 10570855, 15903377, 23898528

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..40.

Michael A. Allen, Combinations without specified separations, Communications in Combinatorics and Optimization (in press).

Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,1,1,1,-1,-1,-1,-1).

FORMULA

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) + a(n-6) + a(n-7) + a(n-8) - a(n-9) - a(n-10) - a(n-11) - a(n-12) for n >= 12.

G.f.: (1 + x + x^2 + 3*x^3 + x^4 + x^5 - x^6 - 3*x^7 - 4*x^8 - 3*x^9 - 2*x^10 - x^11)/(1 - x - x^2 + x^3 - x^4 + x^5 - x^6 - x^7 - x^8 + x^9 + x^10 + x^11 + x^12).

EXAMPLE

For n = 6, the 24 subsets are {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {2,4}, {3,4}, {2,3,4}, {5}, {1,5}, {3,5}, {1,3,5}, {4,5}, {3,4,5}, {6}, {2,6}, {4,6}, {2,4,6}, {5,6}, {4,5,6}.

MATHEMATICA

CoefficientList[Series[(1 + x + x^2 + 3*x^3 + x^4 + x^5 - x^6 - 3*x^7 - 4*x^8 - 3*x^9 - 2*x^10 - x^11)/(1 - x - x^2 + x^3 - x^4 + x^5 - x^6 - x^7 - x^8 + x^9 + x^10 + x^11 + x^12), {x, 0, 38}], x]

LinearRecurrence[{1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1, -1}, {1, 2, 4, 8, 12, 18, 24, 34, 47, 73, 111, 177}, 39]

CROSSREFS

See A375981 for other sequences related to restricted combinations.

Column k=20 of A376033.

KEYWORD

easy,nonn,new

AUTHOR

Michael A. Allen, Sep 20 2024

STATUS

approved

A375979

Number of subsets of {1,2,...,n} such that no two elements differ by 4 or 5.

+0
0

1, 2, 4, 8, 16, 24, 32, 42, 55, 76, 118, 192, 314, 504, 767, 1120, 1612, 2324, 3412, 5148, 7900, 12169, 18631, 28152, 42024, 62364, 92576, 138141, 207629, 313718, 474796, 717456, 1080320, 1620994, 2427447, 3634800, 5450293, 8188936, 12323172, 18555880, 27930853

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..40.

Michael A. Allen, Combinations without specified separations, Communications in Combinatorics and Optimization (in press).

Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,1,2,3,-1,-2,-3).

FORMULA

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) + a(n-7) + 2*a(n-8) + 3*a(n-9) - a(n-10) - 2*a(n-11) - 3*a(n-12) for n >= 12.

G.f.: (1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 6*x^5 + 3*x^6 - x^7 - 3*x^8 - 6*x^9 - 5*x^10 - 3*x^11)/(1 - x - x^3 + x^4 - x^6 - x^7 - 2*x^8 - 3*x^9 + x^10 + 2*x^11 + 3*x^12).

EXAMPLE

For n = 6, the 32 subsets are {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {1,4}, {2,4}, {1,2,4}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, {5}, {2,5}, {3,5}, {2,3,5}, {4,5}, {2,4,5}, {3,4,5}, {2,3,4,5}, {6}, {3,6}, {4,6}, {3,4,6}, {5,6}, {3,5,6}, {4,5,6}, {3,4,5,6}.

MATHEMATICA

CoefficientList[Series[(1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 6*x^5 + 3*x^6 - x^7 - 3*x^8 - 6*x^9 - 5*x^10 - 3*x^11)/(1 - x - x^3 + x^4 - x^6 - x^7 - 2*x^8 - 3*x^9 + x^10 + 2*x^11 + 3*x^12), {x, 0, 38}], x]

LinearRecurrence[{1, 0, 1, -1, 0, 1, 1, 2, 3, -1, -2, -3}, {1, 2, 4, 8, 16, 24, 32, 42, 55, 76, 118, 192}, 39]

CROSSREFS

See A375981 for other sequences related to restricted combinations.

Column k=24 of A376033.

KEYWORD

nonn,easy,new

AUTHOR

Michael A. Allen, Sep 20 2024

STATUS

approved

A376233

G.f. A(x) satisfies: A( x - x^2 - A(x)^3 ) = x.

+0
0

1, 1, 3, 13, 68, 401, 2576, 17670, 127786, 965890, 7583944, 61576198, 515209694, 4430434354, 39073275005, 352812956426, 3257141096415, 30708375193969, 295384611397245, 2896520002777988, 28934822132722103, 294279196616806638, 3045540859482010041, 32057787741226132890

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..300

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) x = A(x - x^2 - A(x)^3).

(2) A(x) = x + A(x)^2 + A(A(x))^3.

(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (x^2 + A(x)^3)^n / n!.

(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (x^2 + A(x)^3)^n/x / n! ).

EXAMPLE

G.f.: A(x) = x + x^2 + 3*x^3 + 13*x^4 + 68*x^5 + 401*x^6 + 2576*x^7 + 17670*x^8 + 127786*x^9 + 965890*x^10 + 7583944*x^11 + 61576198*x^12 + ...

where A(x - x^2 - A(x)^3) = x.

RELATED SERIES.

A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 58*x^6 + 318*x^7 + 1911*x^8 + 12330*x^9 + 84273*x^10 + 604503*x^11 + ...

A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 171*x^6 + 1016*x^7 + 6531*x^8 + 44666*x^9 + 321418*x^10 + ...

A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 230*x^6 + 1560*x^7 + 11139*x^8 + 83120*x^9 + 644472*x^10 + ...

where A(A(x))^3 = A(x) - x - A(x)^2.

A(A(x)) = x + 2*x^2 + 8*x^3 + 42*x^4 + 256*x^5 + 1721*x^6 + 12424*x^7 + 94796*x^8 + 756680*x^9 + ...

where A(A(y)) = x at y = x - 2*x^2 + x^3 - x^4 - (1 - 2*x + 2*x^2)*A(x)^3 - A(x)^6.

PROG

(PARI) {a(n) = my(A=x); if(n<1, 0, for(i=1, n, A = serreverse(x - x^2 - A^3 + x*O(x^n))); polcoeff(A, n))}

for(n=1, 25, print1(a(n), ", "))

(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}

{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (x^2 + A^3)^m)/m! +x*O(x^n))); polcoeff(A, n)}

for(n=1, 25, print1(a(n), ", "))

(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}

{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, (x^2 + A^3)^m/x)/m!) +x*O(x^n))); polcoeff(A, n)}

for(n=1, 25, print1(a(n), ", "))

CROSSREFS

Cf. A211794, A213591.

KEYWORD

nonn,new

AUTHOR

Paul D. Hanna, Sep 20 2024

STATUS

approved

A376321

Obverse convolution (n^2 + 1)**(n^2 + 1); see Comments.

+0
0

2, 9, 144, 5929, 466560, 59213025, 10958689280, 2771535732849, 915539439919104, 382088350057032025, 196357891384811520000, 121752085389995771825625, 89582478947424173216497664, 77138638421388109999960896369, 76829768389915556918132736000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

See A374848 for the definition of obverse convolution and a guide to related sequences.

a(2k+1) is a square for k>=0.

LINKS

Table of n, a(n) for n=0..14.

FORMULA

a(n) ~ n^(2*n+2) / exp(2*n - Pi*n/2). - Vaclav Kotesovec, Sep 20 2024

MATHEMATICA

s[n_] := n^2 + 1; t[n_] := n^2 + 1;

u[n_] := Product[s[k] + t[n - k], {k, 0, n}];

Table[u[n], {n, 0, 20}]

CROSSREFS

Cf. A323540, A374848, A375051.

KEYWORD

nonn,new

AUTHOR

Clark Kimberling, Sep 20 2024

STATUS

approved

A376327

Expansion of e.g.f. -LambertW(-x/(1-x^2)).

+0
0

0, 1, 2, 15, 112, 1285, 17616, 299299, 5946368, 136497897, 3544641280, 102858065431, 3297199331328, 115730076038317, 4414151526557696, 181797547951527915, 8040649885153755136, 380100842138029431121, 19125314442962053300224, 1020539634854353310016415, 57563650890815727219507200

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(n) ~ (1 + 4*exp(-2))^(1/4) * 2^n * n^(n-1) / (exp(n) * (sqrt(4 + exp(2)) - exp(1))^n).

MATHEMATICA

nmax=25; CoefficientList[Series[-LambertW[-x/(1-x^2)], {x, 0, nmax}], x] * Range[0, nmax]!

CROSSREFS

Cf. A052871, A376106.

KEYWORD

nonn,new

AUTHOR

Vaclav Kotesovec, Sep 20 2024

STATUS

approved

A376328

G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^5.

+0
0

1, 5, 40, 380, 3970, 44051, 509575, 6077435, 74194780, 922644310, 11646083631, 148827827450, 1921724362880, 25034267112600, 328614891689845, 4342322118727300, 57715241768897445, 771087466276360970, 10349495416322497575, 139486475071720234920, 1886980259513934080860, 25613816043115261657425

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

Index entries for reversions of series

FORMULA

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^5 ).

G.f.: B(x)^5, where B(x) is the g.f. of A365189.

PROG

(PARI) a(n, s=1, t=5) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^5)/x)

CROSSREFS

Cf. A143927, A365128, A376320.

Cf. A365189.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Sep 20 2024

STATUS

approved

A376320

G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^4.

+0
0

1, 4, 26, 200, 1691, 15180, 142038, 1370076, 13526645, 136024876, 1388394234, 14346699052, 149790104030, 1577765967600, 16745718467070, 178912981116840, 1922688816819276, 20769064846817136, 225384498769815750, 2455985319885345820, 26862562977746930145, 294807644917408047060

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

Index entries for reversions of series

FORMULA

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^4 ).

G.f.: B(x)^4, where B(x) is the g.f. of A365183.

PROG

(PARI) a(n, s=1, t=4) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2)^4)/x)

CROSSREFS

Cf. A143927, A365128, A376328.

Cf. A365183.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Sep 20 2024

STATUS

approved

A376326

Expansion of (1/x) * Series_Reversion( x * (1-x-x^2)^4 ).

+0
0

1, 4, 30, 272, 2737, 29380, 329614, 3818540, 45329440, 548511612, 6740687924, 83898110660, 1055441468145, 13398494365088, 171422870731600, 2208161418665872, 28614197357895055, 372754395074051500, 4878709294080115494, 64123505084010848580, 846018700129069313495

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

Index entries for reversions of series

FORMULA

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(4*n+k+3,k) * binomial(5*n-k+3,n-2*k).

G.f.: B(x)^4, where B(x) is the g.f. of A365188.

PROG

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^2)^4)/x)

(PARI) a(n, s=2, t=4, u=0) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);