A089273 Fifth column (k=6) of array A078739(n,k) ((2,2)-generalized Stirling2).

1, 188, 12052, 540080, 20447056, 706827968, 23178048832, 736079932160, 22912552596736, 704164858293248, 21462936995648512, 650674662791229440, 19656291799888777216, 592413643343696150528, 17826953303927872110592

Offset: 0

Author

Wolfdieter Lang, Nov 07 2003

Keywords

• nonn
• easy

Comments

The numerator of the g.f. is the m=3 row polynomial of the triangle A089275.

References

• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.

Links

• 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 (70, -1708, 17544, -72000, 86400).

Crossrefs

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

Programs

Maple

a:= n-> (Matrix([[12052,188,1,0,0]]). Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [70,-1708,17544, -72000,86400][i] else 0 fi)^n)[1,3]: seq(a(n), n=0..30);  # Alois P. Heinz, Aug 14 2008

Mathematica

LinearRecurrence[{70, -1708, 17544, -72000, 86400}, {1, 188, 12052, 540080, 20447056}, 15] (* Jean-François Alcover, Feb 28 2020 *)

Formula

G.f.: (1+118*x+ 600*x^2)/Product_{p=1..5} (1-(p+1)*p*x).

a(n) = (2^n - 36*6^n + 36*6*12^n - 400*20^n + 75*3*30^n)/6 = d(n) + 118*d(n-1) + 600*d(n-2), n>=2, with d(n) := A089274(n)= A071951(n+5, 5)= (16875*30^n - 20000*20^n + 6048*12^n - 405*6^n + 2*2^n)/2520.