A145438 -id:A145438 - cp's OEIS frontend

A005429 Apéry numbers: n^3*C(2n,n).

0, 2, 48, 540, 4480, 31500, 199584, 1177176, 6589440, 35443980, 184756000, 938929992, 4672781568, 22850118200, 110079950400, 523521630000, 2462025277440, 11465007358860, 52926189069600, 242433164404200, 1102772230560000, 4984806175188840, 22404445765690560

Offset: 0

Simon Plouffe

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
M. Kondratiewa and S. Sadov, Markov's transformation of series and the WZ method, arXiv:math/0405592 [math.CA], 2004.
A. J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
I. J. Zucker, On the series Sum(k>=1) C(2k,k)^(-1)*k^(-n) and related sums, J. Number Theory 20 (1985), no. 1, 92-102.

Cf. A002736, A005258, A005259, A005429, A005430, A145438.

[Binomial(2*n,n)*n^3 : n in [0..30]]; // Wesley Ivan Hurt, Oct 21 2014

Table[n^3 Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Apr 08 2012 *)
CoefficientList[Series[(2*x*(2*x*(2*x+5)+1))/(1-4*x)^(7/2), {x,0,30}], x] (* Vincenzo Librandi, Oct 22 2014 *)

[n^3*binomial(2*n,n) for n in range(31)] # G. C. Greubel, Nov 19 2022

Sum_{n>=1} (-1)^(n+1) / a(n) = 2 * zeta(3) / 5.
G.f.: (2*x*(2*x*(2*x + 5) + 1))/(1-4*x)^(7/2). - Harvey P. Dale, Apr 08 2012
From Ilya Gutkovskiy, Jan 17 2017: (Start)
a(n) ~ 4^n*n^(5/2)/sqrt(Pi).
Sum_{n>=1} 1/a(n) = (1/2)*4F3(1,1,1,1; 3/2,2,2; 1/4) = A145438. (End)

Entry revised by N. J. A. Sloane, Apr 06 2004

A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).

Original entry on oeis.org

5, 0, 5, 4, 2, 9, 4, 7, 4, 6, 8, 3, 5, 1, 9, 2, 4, 1, 6, 4, 2, 4, 5, 0, 4, 8, 1, 9, 0, 8, 4, 3, 2, 1, 4, 9, 1, 8, 8, 6, 6, 9, 0, 1, 4, 5, 6, 8, 2, 6, 2, 8, 6, 4, 9, 8, 2, 6, 6, 4, 7, 1, 2, 8, 7, 5, 7, 3, 3, 4, 7, 3, 3, 7, 6, 1, 7, 5, 9, 0, 6, 8, 2, 7, 1, 6, 4, 5, 3, 3, 1, 8, 1, 5, 0, 0, 1, 3, 6, 6, 1, 9, 6

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.5054294746835192416424504819084321491886690145682628649826647...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)).

S[5] = 2*Pi*Im[PolyLog[4, (-1)^(1/3)]] + (1/9)*Pi^2*Zeta[3] - 19*Zeta[5]/3; RealDigits[S[5], 10, 103] // First

suminf(n=1, 1/(n^5*binomial(2*n,n))) \\ Michel Marcus, Sep 03 2015

S(5) = 2*Pi*Im(PolyLog(4, (-1)^(1/3))) + (1/9)*Pi^2*zeta(3) -19*zeta(5)/3.
Equals (1/2) 4F3(1,1,1,1; 3/2,2,2; 1/4).
Also equals (1/(2592*sqrt(3)))*(Pi*(PolyGamma(3, 1/6) + PolyGamma(3, 1/3) - PolyGamma(3, 2/3) - PolyGamma(3, 5/6))) + (1/9)*Pi^2*zeta(3) - 19*zeta(5)/3.

A261850 Decimal expansion of the central binomial sum S(6), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

5, 0, 2, 6, 7, 6, 5, 2, 1, 4, 7, 8, 2, 6, 9, 2, 8, 6, 4, 5, 4, 6, 7, 7, 4, 5, 9, 9, 7, 9, 3, 4, 8, 6, 3, 9, 6, 6, 4, 6, 0, 2, 6, 0, 0, 0, 9, 1, 6, 4, 0, 6, 6, 1, 4, 6, 8, 6, 2, 7, 6, 5, 2, 3, 2, 4, 8, 7, 1, 6, 1, 5, 0, 8, 8, 5, 4, 6, 3, 1, 2, 1, 1, 7, 6, 2, 3, 4, 1, 5, 7, 2, 7, 8, 4, 0, 5, 2, 7, 6, 7, 8, 5, 4, 1

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.50267652147826928645467745997934863966460260009164...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261851 (S(7)), A261852 (S(8)).

S[6] = Sum[1/(n^6*Binomial[2n, n]), {n, 1, Infinity}]; RealDigits[S[6], 10, 105]//First

Equals (1/2) 7F6(1,1,1,1,1,1,1; 3/2,2,2,2,2,2; 1/4).

Also equals (2/3)*Integral_{0..Pi/3} t*log(2*sin(t/2))^4 dt.

A261851 Decimal expansion of the central binomial sum S(7), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

5, 0, 1, 3, 2, 5, 8, 7, 2, 6, 8, 8, 1, 7, 8, 8, 0, 9, 4, 0, 2, 2, 9, 6, 7, 1, 0, 5, 5, 2, 7, 4, 9, 4, 4, 3, 7, 2, 6, 8, 7, 8, 3, 2, 9, 8, 5, 8, 0, 4, 5, 6, 8, 1, 5, 3, 6, 4, 5, 1, 2, 1, 7, 3, 3, 8, 8, 8, 7, 4, 1, 5, 8, 4, 5, 0, 6, 0, 6, 5, 3, 3, 0, 9, 0, 3, 1, 1, 3, 8, 8, 9, 7, 9, 4, 3, 9, 8, 9, 6, 1, 8, 1, 9, 4

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.501325872688178809402296710552749443726878329858...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261852 (S(8)).

S[7]=-6*Pi*Im[-PolyLog[6, (-1)^(1/3)]] + (17*Pi^4*Zeta[3])/1620 + (1/3)*Pi^2*Zeta[5] - (493*Zeta[7])/24; RealDigits[S[7], 10, 105]//First

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals -6*Pi*Im(-PolyLog(6, (-1)^(1/3))) + (17*Pi^4*zeta(3))/1620 + (1/3)*Pi^2*zeta(5) - (493*zeta(7))/24.

A261852 Decimal expansion of the central binomial sum S(8), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

Original entry on oeis.org

5, 0, 0, 6, 5, 8, 8, 9, 1, 2, 9, 7, 6, 7, 0, 5, 4, 3, 3, 1, 4, 5, 5, 7, 1, 2, 7, 0, 8, 2, 9, 8, 6, 8, 3, 8, 3, 8, 4, 0, 7, 3, 2, 5, 2, 3, 4, 0, 4, 5, 4, 0, 3, 8, 8, 8, 8, 6, 4, 3, 8, 0, 4, 7, 6, 6, 2, 1, 7, 1, 8, 2, 0, 3, 3, 4, 1, 3, 5, 8, 7, 6, 5, 4, 5, 6, 6, 2, 7, 0, 9, 0, 8, 1, 5, 1, 6, 7, 7, 2

Offset: 0

Jean-François Alcover, Sep 03 2015

			0.5006588912976705433145571270829868383840732523404540388886438...

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values, arXiv:hep-th/0004153, 2000.
Eric Weisstein's MathWorld, Central Binomial Coefficient

Cf. A073010 (S(1)), A086463 (S(2)), A145438 (S(3)), A086464 (S(4)), A261839 (S(5)), A261850 (S(6)), A261851 (S(7)).

S[8] = Sum[1/(n^8*Binomial[2n, n]), {n, 1, Infinity}]; RealDigits[S[8], 10, 100] // First

Equals (1/2) 8F7(1,...,1; 3/2,2,...,2; 1/4).

Also equals (4/45)*Integral_{0..Pi/3} t*log(2*sin(t/2))^6 dt.

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A005429 Apéry numbers: n^3*C(2n,n).

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A261839 Decimal expansion of the central binomial sum S(5), where S(k) = Sum_{n>=1} 1/(n^k*binomial(2n,n)).

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A261850 Decimal expansion of the central binomial sum S(6), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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A261851 Decimal expansion of the central binomial sum S(7), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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A261852 Decimal expansion of the central binomial sum S(8), where S(k) = Sum_{n>=1} 1/(n^k binomial(2n,n)).

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