A249908 G.f. (1-x)/(2*sqrt(5*x^2 + 2*x + 1)) - 1/2.

-1, 0, 3, -5, -3, 26, -35, -48, 245, -248, -639, 2355, -1573, -7890, 22555, -6685, -93075, 212280, 27625, -1061415, 1938855, 1276550, -11763465, 16906450, 23324507, -126971664, 136840575, 343314517, -1334857995, 965192298

Offset

1,3

Links

Table of n, a(n) for n=1..30.

Formula

a(n) = Sum_{k=ceiling(n/2)..n} (-1)^k*binomial(k,n-k)*binomial(n-1,n-k).

a(n) = (-1)^k*binomial(n-1, n-k)*binomial(k, n-k)*hypergeom([1, 1+k, -n+k, -n+k], [k, 1/2-n/2+k, 1-n/2+k], -1/4) where k = ceiling(n/2). - Peter Luschny, Nov 09 2014

D-finite with recurrence: n*a(n) + (n+1)*a(n-1) + 3*n*a(n-2) + 5*(-n+3)*a(n-3) = 0. - R. J. Mathar, May 22 2019

n*(4*n-7)*a(n) + 2*(4*n-1)*(n-2)*a(n-1) + 5*(4*n-3)*(n-2)*a(n-2) = 0. - R. J. Mathar, May 22 2019

Maple

T := (n, k) -> (-1)^k*binomial(n-1, n-k)*binomial(k, n-k)*
hypergeom([1, 1+k, -n+k, -n+k], [k, 1/2-n/2+k, 1-n/2+k], -1/4):
seq(simplify(T(n, ceil(n/2))), n=1..30); # Peter Luschny, Nov 09 2014

Mathematica

Rest[CoefficientList[Series[(1 - x) / (2 Sqrt[5 x^2 + 2 x + 1]) - 1/2, {x, 0, 40}], x]] (* Vincenzo Librandi, Nov 08 2014 *)

Prog

(Maxima) a(n):=sum((-1)^k*binomial(k, n-k)*binomial(n-1, n-k), k, ceiling(n/2), n);
(PARI) a(n) = sum(k=ceil(n/2), n, (-1)^k*binomial(k, n-k)*binomial(n-1, n-k)); \\ Michel Marcus, Nov 09 2014

Crossrefs

Sequence in context: A134771 A080349 A249012 * A195418 A366857 A065974

Adjacent sequences: A249905 A249906 A249907 * A249909 A249910 A249911

Keyword

sign

Author

Vladimir Kruchinin, Nov 08 2014

Status

approved