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 A005429 Apéry numbers: n^3*C(2n,n). (Formerly M2169) 7
 0, 2, 48, 540, 4480, 31500, 199584, 1177176, 6589440, 35443980, 184756000, 938929992, 4672781568, 22850118200, 110079950400, 523521630000, 2462025277440, 11465007358860, 52926189069600, 242433164404200, 1102772230560000, 4984806175188840, 22404445765690560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES 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). LINKS 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. FORMULA 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) MAPLE with(combinat):for n from 0 to 22 do printf(`%d, `, n^2*sum(binomial(2*n, n), k=1..n)) od: # Zerinvary Lajos, Mar 13 2007 MATHEMATICA 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, 20}], x] (* Vincenzo Librandi, Oct 22 2014 *) PROG (MAGMA) [Binomial(2*n, n)*n^3 : n in [0..20]]; // Wesley Ivan Hurt, Oct 21 2014 CROSSREFS Cf. A002736, A005258, A005259, A005429, A005430, A145438. Sequence in context: A058090 A051252 A231654 * A035606 A157057 A290690 Adjacent sequences:  A005426 A005427 A005428 * A005430 A005431 A005432 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Entry revised by N. J. A. Sloane, Apr 06 2004 STATUS approved

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Last modified August 9 13:39 EDT 2020. Contains 336323 sequences. (Running on oeis4.)