A307862 Coefficient of x^n in (1 + x - n*x^2)^n.

1, 1, -3, -17, 49, 651, -1259, -38023, 26433, 2969299, 2225101, -289389891, -692529551, 33718183045, 143578976997, -4559187616649, -29119975483135, 699788001188403, 6188699469443869, -119828491083854707, -1404529670244379599, 22563726025297759345, 341997845736800473397

Offset: 0

Author

Seiichi Manyama, May 02 2019

Keywords

• sign

Comments

Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1+4*n)*x^2).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Main diagonal of A307860.

Cf. A002426, A187018, A307885.

Programs

Maple

A307862:= n -> simplify(hypergeom([-n/2, (1-n)/2], [1], -4*n));
seq(A307862(n), n = 0..30); # G. C. Greubel, May 31 2020

Mathematica

a[n_]:= SeriesCoefficient[(1 +x -n*x^2)^n, {x,0,n}]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 31 2020 *)

PARI

{a(n) = polcoef((1+x-n*x^2)^n, n)}

PARI

{a(n) = sum(k=0, n\2, (-n)^k*binomial(n, k)*binomial(n-k, k))}

PARI

{a(n) = sum(k=0, n\2, (-n)^k*binomial(n, 2*k)*binomial(2*k, k))}

Sage

[ hypergeometric([-n/2, (1-n)/2], [1], -4*n).simplify_hypergeometric() for n in (0..30)] # G. C. Greubel, May 31 2020

Formula

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

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

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

a(n) = Hypergeometric2F1(-n/2, (1-n)/2; 1; -4*n). - G. C. Greubel, May 31 2020