A118589 - OEIS

%I #39 Dec 17 2024 18:17:08

%S 1,1,3,13,49,261,1531,9073,63393,465769,3566611,29998101,262167313,

%T 2394499693,23249961099,233439305401,2439472944961,26649502709073,

%U 300078056044963,3498896317045789,42244252226263281,524289088799352661

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

%C Equals row sums of triangle A118588.

%H Seiichi Manyama, <a href="/A118589/b118589.txt">Table of n, a(n) for n = 0..575</a> (terms 0..200 from Vincenzo Librandi)

%F 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

%F 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

%F 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

%F 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

%t Range[0, 21]!*CoefficientList[ Series[ Exp[x*(1-x^3)/(1 - x)], {x, 0, 21}], x] (* _Zerinvary Lajos_, Mar 23 2007 *)

%t With[{nn=30},CoefficientList[Series[Exp[x+x^2+x^3],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 17 2024 *)

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

%o (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 */

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 08 2006