A252355 a(n) = sum_{k = 0..n-1} (-1)^k*C(2*n-1,k)*C(n-1,k), n>0.

1, -2, 1, 8, -29, 34, 92, -512, 919, 818, -9151, 22472, -2924, -156872, 513736, -443392, -2457281, 11094658, -16502221, -31859752, 226433243, -475853006, -217535264, 4333621888, -12126499804, 5346234424

Offset: 1

Author

L. Edson Jeffery, Dec 17 2014

Keywords

• sign
• easy

Comments

For each n > 0, a(n) is an integer such that A234839(p-n) == 2^(2 - 3*n)*a(n) (mod p), for all primes p >= 2*n+1 [Chamberland, et al., Thm. 2.3].

Links

• 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.

Crossrefs

Cf. A000040 (primes), A234839, A045721.

Programs

Mathematica

a[n_] := Sum[(-1)^k*Binomial[2*n - 1, k]*Binomial[n - 1, k], {k, 0, n - 1}]; Table[a[n], {n, 26}]

PARI

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

Formula

a(n) = _2F_1(1-2*n,1-n;1;-1), n>0.

Recurrence: 2*(n-1)*(2*n-1)*(7*n-11)*a(n) = -(91*n^3 - 325*n^2 + 368*n - 128)*a(n-1) - 16*(n-2)*(2*n-3)*(7*n-4)*a(n-2). - Vaclav Kotesovec, Dec 17 2014

Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(2). - Vaclav Kotesovec, Dec 17 2014

exp( Sum_{n >= 1} 2*a(n)*x^n/n ) = 1 + 2*x - 2*x^3 + 4*x^4 - 2*x^5 - 12*x^6 + 40*x^7 - 44*x^8 - 98*x^9 + 520*x^10 - 882*x^11 - 640*x^12 + ... appears to have integer coefficients. - Peter Bala, Jan 04 2016