Displaying 41-50 of 456 results found.
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Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^3 * x^k/k), where phi is the Euler totient function A000010.
+0
3
1, 1, 2, 20, 122, 2122, 15532, 284104, 3837500, 52963964, 1125315224, 20981180464, 500475045688, 10373180665720, 264908485440848, 6624880728277088, 185812008437953808, 5449866267968244496, 167510440639938875680, 5447433174773217714496, 177500241844579492474016
FORMULA
log(a(n)/n!) ~ 2^(9/4) * c^(1/4) * n^(3/4) / 3^(3/4), where c = Product_{p primes} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.337187873791589971961692816152158244949154127758...
MATHEMATICA
nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^3*a[n-k], {k, 1, n}]/n]; Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^3 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^4 * x^k/k), where phi is the Euler totient function A000010.
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3
1, 1, 2, 36, 234, 7290, 54540, 1408680, 23119740, 341788860, 11790437400, 231972879600, 8206299070200, 191673262380600, 6154270418696400, 206515993375692000, 6574758436640394000, 269828090984990538000, 9531096165082736244000, 411037724983993923816000
FORMULA
log(a(n)/n!) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956...
MATHEMATICA
nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^4 * a[n-k], {k, 1, n}]/n]; Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^4 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^4.
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3
1, 4, 60, 1548, 58456, 2930020, 183763704, 13866109012, 1224251041248, 123885272536452, 14140672597851880, 1797709847594145364, 251941291752251706576, 38593132701417704324356, 6415647343472197357272984, 1150373241484390263973203540, 221318733487356013660505462464
FORMULA
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377526. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+3,k)/( (k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+3, k)/((k+1)*(n-k)!));
Decimal expansion of 2*Pi^3/(81*sqrt(3)) + 13*zeta(3)/27.
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3
1, 0, 2, 0, 7, 8, 0, 0, 4, 4, 4, 3, 3, 3, 6, 3, 1, 0, 2, 8, 2, 3, 2, 5, 4, 7, 3, 9, 9, 0, 3, 9, 8, 1, 8, 2, 5, 3, 5, 3, 4, 1, 0, 9, 3, 7, 5, 1, 9, 0, 6, 9, 6, 6, 9, 7, 3, 5, 7, 2, 0, 7, 5, 2, 5, 3, 9, 1, 4, 6, 5, 9, 9, 2, 6, 5, 6, 2, 7, 1, 5, 5, 4, 4, 9, 8, 0, 6, 7, 2, 0, 3, 4, 2, 6, 7, 6, 1, 3, 7
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.
FORMULA
Equals Sum_{k>=0} 1/(3*k + 1)^3 (see Finch).
Equals -psi''(1/3)/54 (see Shamos).
EXAMPLE
1.0207800444333631028232547399039818253534109375...
MATHEMATICA
RealDigits[2Pi^3/(81Sqrt[3])+13Zeta[3]/27, 10, 100][[1]]
Decimal expansion of Pi^3/64 + 7*zeta(3)/16.
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3
1, 0, 1, 0, 3, 7, 2, 9, 6, 8, 2, 6, 2, 0, 0, 7, 1, 9, 0, 1, 0, 4, 2, 0, 2, 8, 6, 8, 5, 8, 4, 7, 1, 8, 6, 7, 0, 9, 9, 4, 4, 5, 1, 6, 3, 6, 7, 4, 0, 9, 2, 3, 0, 6, 8, 5, 0, 5, 1, 2, 7, 2, 1, 3, 3, 3, 4, 0, 2, 9, 1, 3, 5, 6, 1, 6, 9, 1, 3, 6, 3, 3, 7, 9, 3, 5, 5, 4, 1, 4, 8, 3, 3, 8, 5, 0, 4, 2, 7, 2
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.
FORMULA
Equals Sum_{k>=0} 1/(4*k + 1)^3 (see Finch).
Equals -psi''(1/4)/128 = -(psi''(1/8) + psi''(5/8))/1024 (see Shamos).
EXAMPLE
1.01037296826200719010420286858471867099445163674...
MATHEMATICA
RealDigits[Pi^3/64+7Zeta[3]/16, 10, 100][[1]]
Decimal expansion of Sum_{k>=0} 1/(5*k + 1)^3.
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3
1, 0, 0, 5, 9, 1, 2, 1, 4, 4, 4, 5, 7, 7, 4, 3, 7, 3, 2, 2, 3, 6, 7, 9, 2, 3, 6, 0, 1, 4, 7, 0, 0, 1, 4, 4, 8, 2, 5, 4, 9, 3, 6, 1, 1, 2, 0, 6, 4, 0, 2, 4, 5, 8, 2, 4, 7, 0, 3, 3, 3, 9, 6, 5, 0, 7, 1, 0, 0, 0, 5, 7, 4, 8, 0, 7, 3, 9, 3, 4, 6, 2, 0, 2, 7, 7, 4, 1, 1, 7, 8, 1, 0, 7, 3, 1, 2, 0, 3, 6
FORMULA
Equals -psi''(1/5)/250 (see Shamos).
EXAMPLE
1.00591214445774373223679236014700144825493611206...
MATHEMATICA
RealDigits[-PolyGamma[2, 1/5]/250, 10, 100][[1]]
Decimal expansion of Pi^3/(36*sqrt(3)) + 91*zeta(3)/216.
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3
1, 0, 0, 3, 6, 8, 5, 5, 1, 5, 3, 4, 7, 9, 5, 2, 6, 9, 7, 0, 6, 3, 2, 3, 0, 1, 3, 7, 0, 2, 4, 8, 6, 0, 5, 7, 3, 1, 5, 2, 7, 2, 7, 8, 4, 3, 5, 9, 3, 8, 9, 3, 3, 2, 7, 8, 6, 6, 5, 7, 9, 0, 8, 5, 3, 1, 5, 3, 9, 2, 7, 3, 2, 7, 3, 6, 5, 8, 9, 1, 5, 9, 3, 9, 5, 6, 2, 5, 8, 3, 4, 8, 5, 8, 4, 6, 1, 0, 4, 0
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.
FORMULA
Equals Sum_{k>=0} 1/(6*k + 1)^3 (see Finch).
Equals -psi''(1/6)/432 (see Shamos).
EXAMPLE
1.00368551534795269706323013702486057315272784359...
MATHEMATICA
RealDigits[Pi^3/(36*Sqrt[3])+91*Zeta[3]/216, 10, 100][[1]]
E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^3.
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3
1, 3, 30, 483, 11100, 334035, 12478698, 558058179, 29104042152, 1735547479587, 116539815603630, 8704631976941043, 716019297815418732, 64326542671867079955, 6267631435921525638738, 658359915933162131600355, 74168964857766293453918928, 8921104769819780822122624323
FORMULA
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364983. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+3,k)/( (k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k+3, k)/((k+1)*(n-k)!));
E.g.f. satisfies A(x) = (1 + x * exp(x*A(x)))^4.
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3
1, 4, 20, 204, 3112, 61220, 1523064, 45456292, 1586426720, 63461164932, 2862300600040, 143766016251044, 7959047336014416, 481550056915454020, 31615435540393172888, 2238661916541220434660, 170070509857455107126464, 13798559748847266924993284, 1190848786811966457102586824
FORMULA
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377581. a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n-4*k+4,k)/( (n-k+1)*(n-k)! ).
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*n-4*k+4, k)/((n-k+1)*(n-k)!));
E.g.f. satisfies A(x) = 1 + log(1 + x*A(x)^3).
+0
2
1, 1, 5, 47, 654, 12084, 278682, 7708056, 248678784, 9168447600, 380274659760, 17524760349216, 888364833282000, 49125202031205936, 2942774373267939168, 189829708902667840320, 13118899353628035596544, 966975804677206274688000
FORMULA
a(n) = (3*n)! * Sum_{k=0..n} Stirling1(n,k)/(3*n-k+1)!.
MATHEMATICA
Table[(3*n)! * Sum[StirlingS1[n, k]/(3*n-k+1)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 07 2023 *)
PROG
(PARI) a(n) = (3*n)!*sum(k=0, n, stirling(n, k, 1)/(3*n-k+1)!);
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