A335819 E.g.f.: exp((3/2) * x * (2 + x)).

1, 3, 12, 54, 270, 1458, 8424, 51516, 331452, 2230740, 15641424, 113846472, 857706408, 6671592216, 53465326560, 440602852752, 3727748253456, 32332181692464, 287111706003648, 2607272929404000, 24187186030419936, 228997933855499808, 2210786521482955392, 21746223198911853504

Offset: 0

Author

Ilya Gutkovskiy, Nov 20 2020

Keywords

• nonn

Crossrefs

Cf. A000085, A000898, A202830, A294119.

Programs

Mathematica

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

PARI

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

Formula

G.f.: 1 / (1 - 3*x - 3*x^2 / (1 - 3*x - 6*x^2 / (1 - 3*x - 9*x^2 / (1 - 3*x - 12*x^2 / (1 - ...))))), a continued fraction.

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

a(n) = Sum_{k=0..n} binomial(n,k) * A000085(k) * A000898(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A202830(k).

a(n) ~ 3^(n/2) * exp(-3/4 + sqrt(3*n) - n/2) * n^(n/2) / sqrt(2). - Vaclav Kotesovec, Aug 09 2021