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A002736 Apéry numbers: a(n) = n^2*C(2n,n).

%I A002736 M2136 N0848 #133 Aug 03 2025 09:03:11
%S A002736 0,2,24,180,1120,6300,33264,168168,823680,3938220,18475600,85357272,
%T A002736 389398464,1757701400,7862853600,34901442000,153876579840,
%U A002736 674412197580,2940343837200,12759640231800,55138611528000,237371722628040,1018383898440480
%N A002736 Apéry numbers: a(n) = n^2*C(2n,n).
%C A002736 Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-1 of B equals -a(n-1). - _T. D. Noe_, May 01 2011
%D A002736 J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 93.
%D A002736 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002736 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002736 T. D. Noe, <a href="/A002736/b002736.txt">Table of n, a(n) for n = 0..200</a>
%H A002736 Kunle Adegoke, Robert Frontczak, and Taras Goy, <a href="https://math.colgate.edu/~integers/w110/w110.pdf">Fibonacci-Catalan Series</a>, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A110.
%H A002736 Paul S. Bruckman, <a href="https://fq.math.ca/Scanned/37-1/elementary37-1.pdf">Problem B-871</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 85.
%H A002736 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A002736 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 (Annotated scans of some selected pages).
%H A002736 Indulis Strazdins, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-1/elementary38-1.pdf">Solution to problem B-871</a>, Fibonacci Quartely, Vol. 38, No. 1 (2000), pp. 86-87.
%H A002736 Hans J. H. Tuenter, <a href="https://arxiv.org/abs/math/0606080">Walking into an absolute sum</a>, arXiv:math/0606080 [math.NT], 2006. Published version on <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-2/tuenter.pdf">Walking into an absolute sum</a>, The Fibonacci Quarterly, Vol. 40, No. 2 (May 2002), pp. 175-180.
%H A002736 Alfred J. van der Poorten, <a href="http://dx.doi.org/10.1007/BF03028234">A proof that Euler missed ... Apery's proof of the irrationality of zeta(3)</a>, Math. Intelligencer, Vol. 1 (1978/1979), pp. 195-203.
%F A002736 G.f.: x*(4*x+2)/((1-4*x)^(5/2)). - _Marco A. Cisneros Guevara_, Jul 25 2011
%F A002736 Sum_{n>=1} 1/a(n) = Pi^2/18 (Euler). - _Benoit Cloitre_, Apr 07 2002
%F A002736 From _Ilya Gutkovskiy_, Jan 17 2017: (Start)
%F A002736 a(n) ~ 4^n*n^(3/2)/sqrt(Pi).
%F A002736 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)^2 = A086467, where phi is the golden ratio. (End)
%F A002736 D-finite with recurrence: (-n+1)*a(n) +2*(n+4)*a(n-1) +4*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Jan 21 2020
%F A002736 a(n) = (2n)!/(Gamma(n))^2. - _Diego Rattaggi_, Mar 30 2020
%F A002736 a(n) = Sum_{k=0..2*n} binomial(2*n,k)*abs(n-k)^3 (Bruckman, 1999; Strazdins, 2000). - _Amiram Eldar_, Jan 12 2022
%F A002736 Sum_{n>=1} x^n/a(n) = 2*arcsin(sqrt(x)/2)^2, for abs(x) < 4 (Adegoke et al., 2022, section 5, p. 10). - _Amiram Eldar_, Dec 07 2024
%F A002736 From _Peter Bala_, Aug 02 2025: (Start)
%F A002736 For n >= 1,
%F A002736 a(n) = 2*n*(2*n-1)/(n-1)^2 * a(n-1) with a(1) = 2 and
%F A002736 1/a(n) = Sum_{k = 0..n} (-1)^(n+k+1) * binomial(n, k)*binomial(n+k, k)/(n+k)^2. (End)
%F A002736 a(n) = 2 * A002544(n-1) for n>=1. - _Alois P. Heinz_, Aug 03 2025
%p A002736 seq(n^2*binomial(2*n,n), n=0..50); # _Robert Israel_, Aug 07 2014
%t A002736 CoefficientList[ Series[x (4 x + 2)/(1 - 4 x)^(5/2), {x, 0, 20}], x] (* _Robert G. Wilson v_, Aug 08 2011 *)
%t A002736 Table[n^2 Binomial[2n,n],{n,0,30}] (* _Harvey P. Dale_, Jun 21 2017 *)
%o A002736 (MuPAD) combinat::catalan(n)*(n+1)*n^2 $ n = 0..36 // _Zerinvary Lajos_, Apr 17 2007
%o A002736 (Magma) [n^2*Binomial(2*n, n): n in [0..30]]; // _Vincenzo Librandi_, Aug 08 2014
%o A002736 (PARI) my(x='x+O('x^100)); concat(0, Vec(x*(4*x+2)/((1-4*x)^(5/2)))) \\ _Altug Alkan_, Mar 21 2016
%o A002736 (PARI) a(n) = n^2*binomial(2*n, n); \\ _Michel Marcus_, Mar 21 2016
%o A002736 (Sage) [n^2*(n+1)*catalan_number(n) for n in (0..30)] # _G. C. Greubel_, Mar 23 2022
%Y A002736 Cf. A000108, A002544, A005258, A005259, A005429, A005430, A086467.
%Y A002736 A diagonal of A331431.
%K A002736 nonn,easy,nice
%O A002736 0,2
%A A002736 _N. J. A. Sloane_