A307844 Constant term in the expansion of (n/x + 1 + n*x)^n.

1, 1, 9, 55, 1729, 19251, 1050841, 16977129, 1322929665, 28017221059, 2839212609001, 74390784295653, 9283240524317761, 289865990675075725, 42976734096778661817, 1557837326400792009751, 267561369300137776050177, 11042876765198762014337235

Offset: 0

Author

Seiichi Manyama, May 01 2019

Keywords

• nonn

Comments

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

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..351

Crossrefs

Main diagonal of A307847.

Cf. A002426, A092366, A186925, A187018.

Programs

Mathematica

Flatten[{1, Table[Sum[(-1)^k * (2*n + 1)^(n-k) * n^k * Binomial[n,k] * Binomial[2*k,k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 02 2019 *)

PARI

{a(n) = polcoef((n/x+1+n*x)^n, 0)}

PARI

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

PARI

{a(n) = sum(k=0, n, (1-2*n)^(n-k)*n^k*binomial(n, k)*binomial(2*k, k))}

PARI

{a(n) = sum(k=0, n, (1+2*n)^(n-k)*(-n)^k*binomial(n, k)*binomial(2*k, k))}

PARI

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

Formula

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

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

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

a(n) ~ (exp(1/2) + (-1)^n * exp(-1/2)) * 2^(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, May 02 2019