A307947 Coefficient of x^n in 1/(n+1) * (1 + x - n*x^2)^(n+1).

1, 1, -1, -8, 9, 201, -89, -8721, -5599, 540595, 1091551, -43580206, -159753527, 4318835223, 24185472039, -506659112324, -3987724324735, 68460979242171, 722884820238847, -10431188525128096, -143862807220127799, 1761112732849258195, 31284729872945906919

Offset

0,4

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 0..594

Formula

a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,2*k) * A000108(k).

a(n) = Hypergeometric2F1(1/2 - n/2, -n/2, 2, -4*n). - Vaclav Kotesovec, May 12 2021

Maple

f:= n -> coeff(1/(n+1)*(1+x-n*x^2)^(n+1), x, n):
map(f, [$0..30]); # Robert Israel, May 08 2019

Mathematica

a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 23, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, -4*n], {n, 0, 20}] (* Vaclav Kotesovec, May 12 2021 *)

Prog

(PARI) {a(n) = polcoef((1+x-n*x^2)^(n+1)/(n+1), n)}
(PARI) {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, k)*binomial(n-k, k)/(k+1))}
(PARI) {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}

Crossrefs

Cf. A000108, A307862, A307906.

Sequence in context: A239584 A041141 A041142 * A175849 A294468 A041605

Adjacent sequences: A307944 A307945 A307946 * A307948 A307949 A307950

Keyword

sign

Author

Seiichi Manyama, May 07 2019

Status

approved