A118589 Expansion of e.g.f. exp(x + x^2 + x^3).

1, 1, 3, 13, 49, 261, 1531, 9073, 63393, 465769, 3566611, 29998101, 262167313, 2394499693, 23249961099, 233439305401, 2439472944961, 26649502709073, 300078056044963, 3498896317045789, 42244252226263281, 524289088799352661

Offset: 0

Author

Paul D. Hanna, May 08 2006

Keywords

• nonn

Comments

Equals row sums of triangle A118588.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..575 (terms 0..200 from Vincenzo Librandi)

Crossrefs

Cf. A118588 (triangle), A119013 (eigenvector).

Programs

Mathematica

Range[0, 21]!*CoefficientList[ Series[ Exp[x*(1-x^3)/(1 - x)], {x, 0, 21}], x] (* Zerinvary Lajos, Mar 23 2007 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2+x^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 17 2024 *)

Maxima

a(n):=n!*sum(sum(binomial(j,n-3*k+2*j)*binomial(k,j),j,0,k)/k!,k,1,n); /* Vladimir Kruchinin, Sep 01 2010 */

PARI

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

Formula

a(n) = n! * Sum_{k=1..n}(Sum_{j=0..k}(binomial(j,n-3*k+2*j)*binomial(k,j))/k!), n>0. - Vladimir Kruchinin, Sep 01 2010

Recurrence equation: a(n) = a(n-1) + 2*(n-1)*a(n-2) + 3*(n-1)*(n-2)*a(n-3) with initial conditions a(0) = a(1) = 1 and a(2) = 3. - Peter Bala, May 14 2012

E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + (1+x+x^2)/(k+1)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013

a(n) ~ 3^(n/3-1/2) * n^(2*n/3) * exp(7/9*(n/3)^(1/3) + (n/3)^(2/3) - 2*n/3 - 14/81) * (1 + 419/(4374*(n/3)^(1/3)) + 16229573/(191318760*(n/3)^(2/3))). - Vaclav Kotesovec, Oct 09 2013