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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)). 34
 2, 6, 54, 564, 6390, 76356, 948276, 12132504, 158984694, 2124923460, 28877309604, 398046897144, 5554209125556, 78328566695736, 1114923122685720, 15999482238880464, 231253045986317814, 3363838379489630916 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS This sequence is s_18 in Cooper's paper. This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017 Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017 LINKS G. C. Greubel, Table of n, a(n) for n = 0..830 (terms 0..254 from Jason Kimberley) S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012). Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5. FORMULA 1/Pi = 2*3^(-5/2) Sum {k>=0} (n a(n)/18^n) [Cooper, equation (42)] = 2*3^(-5/2) Sum {k>=0} (n a(n)/A001027(n)). 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 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 MATHEMATICA 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 *) PROG (MAGMA) s_18 := func where C is Binomial; (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 (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 CROSSREFS 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.) Sequence in context: A327425 A262046 A280982 * A085078 A152543 A279454 Adjacent sequences:  A219689 A219690 A219691 * A219693 A219694 A219695 KEYWORD nonn,easy AUTHOR Jason Kimberley, Nov 25 2012 STATUS approved

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Last modified October 21 01:23 EDT 2020. Contains 337910 sequences. (Running on oeis4.)