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A143583
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Apéry-like numbers: a(n) = (1/C(2n,n))*Sum_{k=0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k).
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37
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1, 12, 164, 2352, 34596, 516912, 7806224, 118803648, 1818757924, 27972399792, 431824158864, 6686855325888, 103814819552016, 1615296581684928, 25180747436810304, 393189646497706752, 6148451986328464164, 96269310864931432368, 1509065592479205772304
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OFFSET
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0,2
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COMMENTS
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These numbers bear some analogy to the Apéry numbers A005258. They appear in the evaluation of the spectral zeta function of the non-commutative harmonic oscillator zeta_Q(s) at s = 2 and satisfy a recurrence relation similar to the one satisfied by the Apéry numbers.
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LINKS
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FORMULA
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a(n) = (1/C(2n,n))*sum {k = 0..n} C(2k,k)*C(4k,2k)*C(2n-2k,n-k)*C(4n-4k,2n-2k).
Recurrence relation:
a(0) = 1, a(1) = 12, n^2*a(n) = 4*(8*n^2-8*n+3)*a(n-1) - 256*(n-1)^2*a(n-2).
Congruences:
For odd prime p, a(m*p^r) = a(m*p^(r-1)) (mod p^r) for any m,r in N.
a(n) ~ 16^n/(Pi*sqrt(Pi*n)) * (log(n) + gamma + 6*log(2)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 11 2013
a(n) = sum {k = 0..n} 4^(n-k) C(2k,k)^2*C(2n-2k,n-k). - Tito Piezas III, Dec 12 2014
a(n) = hypergeom([1/2,1/2,n+1],[1,n+3/2],1)*2^(5*n+1)*n!/((2*n+1)!!*Pi) - G. A. Edgar, Dec 10 2016
a(n) = binomial(4*n,2*n)*hypergeom([1/4,3/4,-n,-n], [1,1/4-n,3/4-n], 1). - Peter Luschny, May 14 2020
a(n) = 16^n*Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2*binomial(n, k).
a(n) = 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1). (End)
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EXAMPLE
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G.f. = 1 + 12*x + 164*x^2 + 2352*x^3 + 34596*x^4 + 516912*x^5 + ...
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MAPLE
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a := n -> 1/binomial(2*n, n)*add(binomial(2*k, k)*binomial(4*k, 2*k)*binomial(2*n-2*k, n-k)*binomial(4*n-4*k, 2*n-2*k), k = 0..n): seq(a(n), n = 0..25);
series( 2*EllipticK(4*x^(1/2))/(Pi*sqrt(1-16*x)), x=0, 20); # Mark van Hoeij, Apr 06 2013
A143583 := n -> 16^n*hypergeom([1/2, 1/2, -n], [1, 1], 1):
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MATHEMATICA
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Table[1/Binomial[2*n, n]*Sum[Binomial[2*k, k]*Binomial[4*k, 2*k]*Binomial[2*n-2*k, n-k]*Binomial[4*n-4*k, 2*n-2*k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 11 2013 *)
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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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easy,nonn
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AUTHOR
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STATUS
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approved
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