A202837 E.g.f.: exp(3*x + 2*x^2).

1, 3, 13, 63, 345, 2043, 13029, 88119, 629169, 4707315, 36772029, 298608687, 2513795337, 21874602987, 196341166485, 1814001266727, 17222473789281, 167763502438371, 1674418724986221, 17102228350521375, 178562508150516921, 1903865792493260763

Offset: 0

Author

Paul D. Hanna, Dec 25 2011

Keywords

• nonn

Examples

			E.g.f.: 1 + 3*x + 13*x^2/2! + 63*x^3/3! + 345*x^4/4! + 2043*x^5/5! +...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..658
• Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, Parabolic orbits of 2-nilpotent elements for classical groups, arXiv:1802.06425 [math.RT], 2018.

Crossrefs

Cf. A202836.

Programs

Mathematica

CoefficientList[Series[E^(3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

{a(n)=n!*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)}

PARI

{a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))}

PARI

/* O.g.f. as a continued fraction: */
{a(n)=local(CF=1+3*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-3*x-4*(n-k)*x^2*CF)); polcoeff(CF, n)}

Formula

a(n) = Sum_{k=0..[n/2]} 3^(n-2*k)*2^k * n!/((n-2*k)!*k!).

O.g.f.: 1/(1-3*x - 4*x^2/(1-3*x - 8*x^2/(1-3*x - 12*x^2/(1-3*x - 16*x^2/(1-3*x -...))))), a continued fraction.

Recurrence: a(n) = 3*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 2^(n-1/2)*exp(3/2*sqrt(n)-n/2-9/16)*n^(n/2)*(1+33/(64*sqrt(n))). - Vaclav Kotesovec, Oct 20 2012