A324591 E.g.f.: exp(2 * (x + x^2 / 2 + x^3 / 3)).

1, 2, 6, 24, 108, 552, 3144, 19392, 129168, 920736, 6958944, 55582848, 466824384, 4104798336, 37688879232, 360236187648, 3575154053376, 36768528142848, 391060780180992, 4293782854170624, 48597548604926976, 566152604314232832, 6780179847538722816, 83375209195856216064

Offset: 0

Author

Ilya Gutkovskiy, Nov 20 2020

Keywords

• nonn

Crossrefs

Cf. A000898, A057693.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Exp[2 (x + x^2/2 + x^3/3)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[1] = 2; a[2] = 6; a[n_] := a[n] = 2 (a[n - 1] + (n - 1) a[n - 2] + (n - 1) (n - 2) a[n - 3]); Table[a[n], {n, 0, 23}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(2*(x + x^2/2 + x^3/3)))) \\ Michel Marcus, Nov 21 2020

Formula

D-finite recurrence: a(n) = 2 * (a(n-1) + (n-1) * a(n-2) + (n-1) * (n-2) * a(n-3)).

a(n) = Sum_{k=0..n} binomial(n,k) * A057693(k) * A057693(n-k).