A118588 -id:A118588 - cp's OEIS frontend

A119013 Eigenvector of triangle A118588; E.g.f. satisfies: A(x) = exp(x)*A(x^2+x^3).

1, 1, 3, 13, 73, 621, 5491, 60313, 743793, 10115353, 158914531, 2815311621, 55094081593, 1142894689093, 25142695616403, 594557634923281, 15084112106943841, 407999468524242993, 11669035487641120963

Offset: 0

Paul D. Hanna, May 08 2006

nonn

E.g.f. of triangle A118588 is exp(x + y*(x^2+x^3)); note the similarity to the e.g.f. of this sequence. More generally, the e.g.f. of an eigenvectors can be determined from the e.g.f. of a triangle as follows. [ Given a triangle with e.g.f.: exp(x + y*x*F(x)) such that F(0) = 0, then the eigenvector has e.g.f.: exp(G(x)) where o.g.f. G(x) satisfies: G(x) = x + G(x*F(x)). ]

			A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 621*x^5/5! +...
log(A(x)) = o.g.f. of A014535 =
x + x^2+ x^3+ x^4+ 2*x^5+ 2*x^6+ 3*x^7+ 4*x^8+ 5*x^9+ 8*x^10 +...

Cf. A118588 (triangle), A118589 (row sums), A014535 (log(A(x))).

{a(n)=if(n==0,1,sum(k=0,n\2,a(k)*n!*polcoeff(polcoeff(exp(x+y*(x^2+x^3)+x*O(x^n)+y*O(y^k)),n,x),k,y)))}

Log(A(x)) = o.g.f. of A014535 (B-trees of order 3 with n leaves).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A119013 Eigenvector of triangle A118588; E.g.f. satisfies: A(x) = exp(x)*A(x^2+x^3).

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