A323770 Expansion of e.g.f. x*(2 - x)*exp(x/(1 - x))/(2*(1 - x)^2).

0, 1, 5, 30, 214, 1775, 16791, 178360, 2101100, 27172269, 382566025, 5823044546, 95253119490, 1666020561595, 31019392831259, 612430207741500, 12778091116288216, 280893425932078745, 6487870112636577165, 157066777096248548134, 3976727555939887035950, 105087648979005066820551

Offset: 0

Author

Ilya Gutkovskiy, Jan 27 2019

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..430

Crossrefs

Cf. A000217, A052852, A103194.

Programs

Magma

[n eq 0 select 0 else (&+[Binomial(n-1,k)*Binomial(k+2,2)* Factorial(n)/Factorial(k+1): k in [0..n-1]]): n in [0..20]]; // G. C. Greubel, Mar 05 2021

Maple

seq(n!*coeff(series(x*(2-x)*exp(x/(1-x))/(2*(1-x)^2),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 29 2019

Mathematica

nmax = 21; CoefficientList[Series[x (2 - x) Exp[x/(1 - x)]/(2 (1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Binomial[k + 1, 2] n!/k!, {k, 0, n}], {n, 0, 21}]

Sage

[0]+[sum(binomial(n-1,k)*binomial(k+2,2)*factorial(n)/factorial(k+1) for k in (0..n-1)) for n in [1..20]] # G. C. Greubel, Mar 05 2021

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000217(k)*n!/k!.

a(n) ~ n^(n + 3/4) / (2^(3/2) * exp(n - 2*sqrt(n) + 1/2)). - Vaclav Kotesovec, Jan 27 2019

a(n) = n!*Hypergeometric2F2([1-n, 3], [1, 2], -1). - G. C. Greubel, Mar 05 2021