A118589 E.g.f.: A(x) = 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

0,3

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)

Formula

a(n) = n!*sum(sum(binomial(j,n-3*k+2*j)*binomial(k,j),j,0,k)/k!,k,1,n), 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

Mathematica

Range[0, 21]!*CoefficientList[ Series[ Exp[x*(1-x^3)/(1 - x)], {x, 0, 21}], x] # Zerinvary Lajos, Mar 23 2007

Prog

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

Crossrefs

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

Sequence in context: A218420 A241775 A045908 * A113237 A284329 A259338

Adjacent sequences: A118586 A118587 A118588 * A118590 A118591 A118592

Keyword

nonn

Author

Paul D. Hanna, May 08 2006

Status

approved