A119013 -id:A119013 - cp's OEIS frontend

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

Paul D. Hanna, May 08 2006

nonn

Equals row sums of triangle A118588.

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

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

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 *)

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 */

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

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

A118588 Triangle generated by e.g.f.: A(x,y) = exp(x + y*(x^2+x^3)), read by rows of length [n/2+1].

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 36, 12, 1, 80, 180, 1, 150, 1260, 120, 1, 252, 5460, 3360, 1, 392, 17640, 43680, 1680, 1, 576, 46872, 342720, 75600, 1, 810, 108360, 1839600, 1587600, 30240, 1, 1100, 225720, 7539840, 20235600, 1995840, 1, 1452, 433620, 25391520

Offset: 0

Paul D. Hanna, May 08 2006

E.g.f. V(x) of eigenvector A119013 satisfies: V(x) = exp(x)*V(x^2+x^3); note the similarity to e.g.f. of this triangle.

			Triangle begins:
1;
1;
1,2;
1,12;
1,36,12;
1,80,180;
1,150,1260,120;
1,252,5460,3360;
1,392,17640,43680,1680;
1,576,46872,342720,75600; ...
O.g.f. for columns:
0!/0!*(1)/(1-x);
2!/1!*(1+2*x)/(1-x)^4;
4!/2!*(1+8*x+21*x^2)/(1-x)^7;
6!/3!*(1+18*x+129*x^2+356*x^3)/(1-x)^10;
8!/4!*(1+32*x+438*x^2+2984*x^3+8425*x^4)/(1-x)^13; ...

Cf. A118589 (row sums), A119013 (eigenvector).

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

cp's OEIS Frontend

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

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