A118395 Expansion of e.g.f. exp(x + x^3).

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 91225, 681121, 3493711, 33597961, 303321877, 1938378625, 20282865331, 211375647841, 1607008257841, 18157826367937, 212200671085975, 1860991143630841, 22560913203079021, 289933758771407521, 2869267483843753147, 37116733726117707025

Offset: 0

Author

Paul D. Hanna, May 07 2006

Keywords

• nonn

Comments

Equals row sums of triangle A118394.

Links

• Robert Israel, Table of n, a(n) for n = 0..533

Crossrefs

Cf. A118394, A118396.

Cf. A047974, A190875, A190877.

Programs

Magma

[n le 3 select 1 else Self(n-1) + 3*(n-2)*(n-3)*Self(n-3): n in [1..26]]; // Vincenzo Librandi, Aug 25 2015

Maple

with(combstruct):seq(count(([S, {S=Set(Union(Z, Prod(Z, Z, Z)))}, labeled], size=n)), n=0..22); # Zerinvary Lajos, Mar 18 2008

Mathematica

CoefficientList[Series[E^(x+x^3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)
T[n_, k_] := n!/(k!(n-3k)!);
a[n_] := Sum[T[n, k], {k, 0, Floor[n/3]}];
a /@ Range[0, 24] (* Jean-François Alcover, Nov 04 2020 *)

PARI

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

PARI

N=33;  x='x+O('x^N);
egf=exp(x+x^3);
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

PARI

a(n) = n!*sum(k=0, n\3, binomial(n-2*k, k)/(n-2*k)!); \\ Seiichi Manyama, Feb 25 2022

Sage

def a(n):
    if (n<3): return 1
    else: return a(n-1) + 3*(n-1)*(n-2)*a(n-3)
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021

Formula

E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + (1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1) )), recursively defined continued fraction. - Sergei N. Gladkovskii, Feb 04 2013

a(n) ~ 3^(n/3-1/2) * n^(2*n/3) * exp((n/3)^(1/3)-2*n/3). - Vaclav Kotesovec, Jun 02 2013

E.g.f.: A(x) = exp(x+x^3) satisfies A' - (1+3*x^2)*A = 0. - Gheorghe Coserea, Aug 24 2015

a(n+1) = a(n) + 3*n*(n-1)*a(n-2). - Gheorghe Coserea, Aug 24 2015

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,k)/(n-2*k)!. - Seiichi Manyama, Feb 25 2022

Extensions

Missing a(0)=1 prepended by Joerg Arndt, Sep 15 2012