A003013 E.g.f. 1 + x*exp(x) + x^2*exp(2*x).

1, 1, 4, 15, 52, 165, 486, 1351, 3592, 9225, 23050, 56331, 135180, 319501, 745486, 1720335, 3932176, 8912913, 20054034, 44826643, 99614740, 220200981, 484442134, 1061158935, 2315255832, 5033164825

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-25,38,-28,8).

Crossrefs

Cf. A048482, A001787, A005183.

Cf. A002999, A001815.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[1+x Exp[x]+x^2 Exp[2x],{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},LinearRecurrence[{8,-25,38,-28,8},{1,4,15,52,165},30]] (* Harvey P. Dale, Nov 01 2011 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 8,-28,38,-25,8]^n*[1;1;4;15;52])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

Formula

From Ralf Stephan, Sep 02 2003: (Start)

a(0) = 1, a(n) = (n^2 - n)*2^n/4 + n.

a(n) = A002999(n) - n = A001815(n) + n. (End)

O.g.f.: 1+x*(-1+4*x-8*x^2+6*x^3) / ( (x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Mar 22 2011

a(n) = 8*a(n-1) - 25*a(n-2) + 38*a(n-3) - 28*a(n-4) + 8*a(n-5); a(0)=1, a(1)=1, a(2)=4, a(3)=15, a(4)=52, a(5)=165. - Harvey P. Dale, Nov 01 2011