A301741 a(n) = n! * [x^n] exp((n + 1)*x + x^2/2).

1, 2, 10, 76, 778, 10026, 155884, 2839880, 59339004, 1399069450, 36746349496, 1064024248068, 33676500286840, 1156685567791586, 42850609041047760, 1703182952266379536, 72299420602524921616, 3264579136056004359570, 156238968782480840396704, 7900247992586138688381500

Offset: 0

Author

Ilya Gutkovskiy, Mar 26 2018

Keywords

• nonn

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..200
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000085, A005425, A202834, A202879.

Programs

GAP

List([0..10],n->Sum([0..Int(n/2)],k->Factorial(n)*(n+1)^(n-2*k)/(2^k*Factorial(k)*Factorial(n-2*k)))); # Muniru A Asiru, Mar 26 2018

Mathematica

Table[n! SeriesCoefficient[Exp[(n + 1) x + x^2/2], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - (n + 1) x + ContinuedFractionK[-k x^2, 1 - (n + 1) x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[Sum[n! (n + 1)^(n - 2 k)/(2^k k! (n - 2 k)!), {k, 0, Floor[n/2]}], {n, 0, 19}]

Formula

a(n) = [x^n] 1/(1 - (n + 1)*x - x^2/(1 - (n + 1)*x - 2*x^2/(1 - (n + 1)*x - 3*x^2/(1 - ...)))), a continued fraction.

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

a(n) ~ exp(3/2) * n^n. - Vaclav Kotesovec, Apr 08 2018