A307885 Coefficient of x^n in (1 - (n-1)*x - n*x^2)^n.

1, 0, -3, 28, -255, 2376, -20195, 71688, 3834369, -187855280, 6676401501, -220595216280, 7180102389889, -234023553073296, 7631745228481725, -245429882267144624, 7501602903392006145, -196609711096827812448, 2542435002501531333949

Offset

0,3

Comments

Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*(n-1)*x + ((n+1)*x)^2).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..386

Formula

a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^2.

a(n) = Sum_{k=0..n} (-n-1)^(n-k) * binomial(n,k) * binomial(n+k,k).

a(n) = Hypergeometric2F1(-n, -n, 1, -n). - Vaclav Kotesovec, May 07 2019

a(n) = n! * [x^n] exp((1 - n)*x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020

Maple

A307885:= n -> simplify(hypergeom([-n, -n], [1], -n));
seq(A307885(n), n = 0..30); # G. C. Greubel, May 31 2020

Mathematica

Table[Hypergeometric2F1[-n, -n, 1, -n], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2019 *)

Prog

(PARI) {a(n) = polcoef((1-(n-1)*x-n*x^2)^n, n)}
(PARI) {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^2)}
(PARI) {a(n) = sum(k=0, n, (-n-1)^(n-k)*binomial(n, k)*binomial(n+k, k))}
(Sage) [ hypergeometric([-n, -n], [1], -n).simplify_hypergeometric() for n in (0..30)] # G. C. Greubel, May 31 2020

Crossrefs

Main diagonal of A307884.

Cf. A187021.

Sequence in context: A037665 A037784 A037588 * A076723 A198887 A026114

Adjacent sequences: A307882 A307883 A307884 * A307886 A307887 A307888

Keyword

sign

Author

Seiichi Manyama, May 02 2019

Status

approved