A097814 E.g.f. exp(3x)/(1-3x).

1, 6, 45, 432, 5265, 79218, 1426653, 29961900, 719092161, 19415508030, 582465299949, 19221355075464, 691968783248145, 26986782548271978, 1133444867032206045, 51005019016463620932, 2448240912790296851457

Offset: 0

Author

Paul Barry, Aug 26 2004

Keywords

• easy
• nonn

Links

• Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Crossrefs

Cf. A082032.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Exp[3x]/(1-3x),{x,0,nn}],x] Range[ 0,nn]!] (* or *) RecurrenceTable[{a[0]==1,a[n]==3n a[n-1]+3^n},a,{n,20}] (* Harvey P. Dale, Feb 23 2012 *)

Formula

a(n) = 3n*a(n-1)+3^n, n>0, a(0)=1; a(n) = 3^n*A000522(n).

G.f.: 1/Q(0), where Q(k) = 1 - 6*x*(k+1) - 9*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013

Conjecture: a(n) +3*(-n-1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Dec 21 2014