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A219692
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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)).
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34
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2, 6, 54, 564, 6390, 76356, 948276, 12132504, 158984694, 2124923460, 28877309604, 398046897144, 5554209125556, 78328566695736, 1114923122685720, 15999482238880464, 231253045986317814, 3363838379489630916
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OFFSET
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0,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
a(n) ~ 3 * 2^(4*n + 1/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023
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MATHEMATICA
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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 *)
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PROG
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(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;
(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
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CROSSREFS
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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.)
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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