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A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

%I A219692 #43 Mar 08 2023 03:54:49
%S A219692 2,6,54,564,6390,76356,948276,12132504,158984694,2124923460,
%T A219692 28877309604,398046897144,5554209125556,78328566695736,
%U A219692 1114923122685720,15999482238880464,231253045986317814,3363838379489630916
%N A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).
%C A219692 This sequence is s_18 in Cooper's paper.
%C A219692 This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A219692 Every prime eventually divides some term of this sequence. - _Amita Malik_, Aug 20 2017
%H A219692 G. C. Greubel, <a href="/A219692/b219692.txt">Table of n, a(n) for n = 0..830</a> (terms 0..254 from Jason Kimberley)
%H A219692 S. Cooper, <a href="http://dx.doi.org/10.1007/s11139-011-9357-3">Sporadic sequences, modular forms and new series for 1/pi</a>, Ramanujan J. (2012).
%H A219692 Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See s18 p. 3.
%H A219692 Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5.
%F A219692 1/Pi
%F A219692 = 2*3^(-5/2) Sum {k>=0} (n a(n)/18^n) [Cooper, equation (42)]
%F A219692 = 2*3^(-5/2) Sum {k>=0} (n a(n)/A001027(n)).
%F A219692 G.f.: 1+hypergeom([1/8, 3/8],[1],256*x^3/(1-12*x)^2)^2/sqrt(1-12*x). - _Mark van Hoeij_, May 07 2013
%F A219692 Conjecture D-finite with recurrence: n^3*a(n) -2*(2*n-1)*(7*n^2-7*n+3)*a(n-1) +12*(4*n-5)*(n-1)* (4*n-3)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016
%F A219692 a(n) ~ 3 * 2^(4*n + 1/2) / (Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2023
%t A219692 Table[Sum[(-1)^j*Binomial[n,j]*Binomial[2j,j]*Binomial[2n-2j, n-j]* (Binomial[2n-3j-1,n] +Binomial[2n-3j,n]), {j,0,Floor[n/3]}], {n,0,20}] (* _G. C. Greubel_, Oct 24 2017 *)
%o A219692 (Magma) s_18 := func<k|&+[(-1)^j*C(k,j)*C(2*j,j)*C(2*k-2*j,k-j)*(C(2*k-3*j-1,k)+C(2*k-3*j,k)):j in[0..k div 3]]> where C is Binomial;
%o A219692 (PARI) {a(n) = sum(j=0,floor(n/3), (-1)^j*binomial(n,j)*binomial(2*j,j)* binomial(2*n-2*j,n-j)*(binomial(2*n-3*j-1,n) +binomial(2*n-3*j,n)))}; \\ _G. C. Greubel_, Apr 02 2019
%o A219692 (Sage) [sum((-1)^j*binomial(n,j)*binomial(2*j,j)*binomial(2*n-2*j,n-j)* (binomial(2*n-3*j-1,n)+binomial(2*n-3*j,n)) for j in (0..floor(n/3))) for n in (0..20)] # _G. C. Greubel_, Apr 02 2019
%Y A219692 The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%K A219692 nonn,easy
%O A219692 0,1
%A A219692 _Jason Kimberley_, Nov 25 2012