A295183 a(n) = n! * [x^n] exp(n*x)/(1 - x)^n.

1, 2, 18, 276, 5960, 165870, 5648832, 227507336, 10577029248, 557457222330, 32843470246400, 2139014862736092, 152592485390272512, 11833139429253625574, 991101777088623943680, 89164680959505831930000, 8575295241502192869343232, 877955050581430468997781234, 95337079570413427211596726272

Offset: 0

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

• nonn

Comments

The n-th term of the n-fold exponential convolution of A000522 with themselves.

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Laguerre Polynomial
• Wikipedia, Laguerre polynomials
• Index entries for sequences related to factorial numbers
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A000407, A000522, A001813, A081923, A295182, A295188.

Programs

Mathematica

Table[n! SeriesCoefficient[Exp[n x]/(1 - x)^n, {x, 0, n}], {n, 0, 18}]

Formula

a(n) ~ phi^(3*n + 1/2) * n^n / (5^(1/4) * exp(n/phi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017

a(n) = (-1)^n*n!*Laguerre(n,-2*n,n). - Ilya Gutkovskiy, May 24 2018