A089273 - OEIS

%I #19 Aug 25 2024 17:34:41

%S 1,188,12052,540080,20447056,706827968,23178048832,736079932160,

%T 22912552596736,704164858293248,21462936995648512,650674662791229440,

%U 19656291799888777216,592413643343696150528,17826953303927872110592

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

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

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

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="https://arxiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (70, -1708, 17544, -72000, 86400).

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

%F 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.

%p 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

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

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

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 07 2003