A089271 Third column (k=4) of array A078739(n,k) ((2,2)-generalized Stirling2).

1, 38, 652, 9080, 116656, 1446368, 17636032, 213311360, 2569812736, 30898216448, 371141389312, 4455873443840, 53483541999616, 641880868118528, 7703040602324992, 92439308337643520, 1109288626710839296

Offset: 0

Author

Wolfdieter Lang, Nov 07 2003

Keywords

• nonn
• easy

Comments

The numerator of the g.f. is the n=2 row polynomial of the triangle A089275.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• Index entries for linear recurrences with constant coefficients, signature (20, -108, 144).

Crossrefs

Cf. A089272, A071951 (Legendre-Stirling triangle).

Programs

Magma

[6*12^n-6*6^n+2^n: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Mathematica

Table[6*12^n -6*6^n +2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{20,-108,144},{1,38,652},20] (* Harvey P. Dale, Oct 22 2024 *)

PARI

for(n=0,30, print1(6*12^n -6*6^n +2^n, ", ")) \\ G. C. Greubel, Feb 07 2018

Formula

G.f.: (1+18*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)).

a(n) = 6*12^n - 6*6^n + 2^n = d(n) + 18*d(n-1), n>=1, a(0)=1, with d(n) := A016309(n) = A071951(n+3, 3) = (24*12^n-15*6^n+2^n)/10.