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A234839 a(n) = Sum_{k = 0..n} (-1)^k * C(n,k) * C(2*n,k). 4
1, -1, -1, 8, -17, -1, 116, -344, 239, 1709, -7001, 9316, 22276, -138412, 268568, 189008, -2608913, 6809417, -1814851, -45852416, 159116983, -155628353, -720492928, 3481793888, -5558713852, -9029921876, 71541001076, -158672882224, -45300345128, 1370202238072 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

For each n > 0, a(p-n) == 2^(2 - 3*n)*A252355(n) (mod p), for all primes p >= 2*n+1 [Chamberland, et al., Thm. 2.3]. - L. Edson Jeffery, Dec 17 2014

LINKS

Table of n, a(n) for n=0..29.

Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672.

FORMULA

Recurrence: 2*n*(2*n-1)*(7*n-10)*a(n) =  -(91*n^3 - 221*n^2 + 160*n - 36)*a(n-1) - 16*(n-1)*(2*n-3)*(7*n-3)*a(n-2).

Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(2).

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 - x + 3*x^3 - 7*x^4 + 4*x^5 + 24*x^6 - 85*x^7 + 99*x^8 + 215*x^9 - 1196*x^10 + ... appears to have integer coefficients. - Peter Bala, Jan 04 2016

MATHEMATICA

Table[Sum[(-1)^k*Binomial[n, k]*Binomial[2*n, k], {k, 0, n}], {n, 0, 20}]

Table[Hypergeometric2F1[-2*n, -n, 1, -1], {n, 0, 20}]

PROG

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2*n, k)); \\ Michel Marcus, Jan 13 2016

CROSSREFS

Cf. A005809, A252355.

Sequence in context: A024280 A115434 A024107 * A066554 A302976 A244537

Adjacent sequences:  A234836 A234837 A234838 * A234840 A234841 A234842

KEYWORD

sign,easy

AUTHOR

Vaclav Kotesovec, Dec 31 2013

STATUS

approved

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Last modified November 18 19:59 EST 2019. Contains 329288 sequences. (Running on oeis4.)