A331688 E.g.f.: exp(-x/(1 - x)) / (1 - 2*x).

1, 1, 3, 17, 137, 1389, 16819, 236557, 3792753, 68326073, 1366917731, 30074632521, 721798881913, 18766625660197, 525460685327187, 15763716503597189, 504436925448024929, 17150818356045629937, 617428780939911647683, 23462281235407345160833

Offset: 0

Author

Ilya Gutkovskiy, Jan 24 2020

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..403

Crossrefs

Cf. A000165, A000166, A000354, A009940, A293116, A331689.

Programs

Maple

f:= gfun:-rectoproc({a(n) = -(n - 1)*(5*n - 8)*a(n - 2) + (-3 + 4*n)*a(n - 1) + 2*(n - 1)*(n - 2)^2*a(n - 3),a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 28 2020

Mathematica

nmax = 19; CoefficientList[Series[Exp[-x/(1 - x)]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! Subfactorial[n - k], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000166(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * k! * 2^k * A293116(n-k).

a(n) ~ n! * exp(-1) * 2^n. - Vaclav Kotesovec, Jan 26 2020

a(n) = (4*n-3)*a(n-1)-(n-1)*(5*n-8)*a(n-2)+2*(n-1)*(n--2)^2*a(n-3). - Robert Israel, Jul 28 2020