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A143583
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Apery-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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1, 12, 164, 2352, 34596, 516912, 7806224, 118803648, 1818757924, 27972399792, 431824158864, 6686855325888, 103814819552016, 1615296581684928, 25180747436810304, 393189646497706752, 6148451986328464164
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| These numbers bear some analogy to the Apery 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 Apery numbers.
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LINKS
| K. Kimoto and M. Wakayama, Apery-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators Kyushu J. Math. Vol. 60, 2006, 383-404.
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FORMULA
| 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.
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MAPLE
| a := n -> 1/binomial(2n, n)*add(binomial(2k, k)*binomial(4k, 2k)*binomial(2n-2k, n-k)*binomial(4n-4k, 2n-2k), k = 0..n): seq(a(n), n = 0..20):
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CROSSREFS
| Cf. A005258.
Sequence in context: A138455 A024221 A093152 * A203372 A193104 A046174
Adjacent sequences: A143580 A143581 A143582 * A143584 A143585 A143586
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KEYWORD
| easy,nonn
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008
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