A307906 Coefficient of x^n in 1/(n+1) * (1 + x + n*x^2)^(n+1).

1, 1, 3, 10, 57, 301, 2251, 15583, 138209, 1153603, 11592451, 111381348, 1235739385, 13276480803, 159935056555, 1884023828326, 24356065951617, 310189106485419, 4266048524240323, 58124516559463590, 844705360693479801, 12213285476055278959, 186543178982826381387

Offset: 0

Author

Seiichi Manyama, May 05 2019

Keywords

• nonn

Comments

Also coefficient of x^n in the expansion of 2/(1 - x + 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 A306684.

Cf. A000108, A001006, A187018, A247496, A292716.

Programs

Mathematica

Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, 4*n], {n, 0, 20}] (* Vaclav Kotesovec, May 05 2019 *)

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))}

Formula

a(n) = [x^n] (1 - x - sqrt(1 - 2*x + (1 - 4*n)*x^2))/(2*n*x^2).

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) ~ exp(sqrt(n)/2 - 1/8) * 2^(n + 1/2) * n^((n-3)/2) / sqrt(Pi). - Vaclav Kotesovec, May 05 2019