Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Aug 28 2019 17:08:07
%S 1,30,12,1,2700,1920,426,36,1,491400,478800,162540,25344,1956,72,1,
%T 150368400,181440000,80451000,17624880,2130660,147840,5820,120,1,
%U 69470200800,98424849600,52905560400,14618016000,2346624000,232202880
%N Generalized Stirling2 array (6,2).
%C The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.
%D P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
%D M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
%H W. Lang, <a href="/A091746/a091746.txt">First 6 rows</a>.
%F a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+4*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=6, s=2.
%F Recursion: a(n, k)=sum(binomial(2, p)*fallfac(4*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
%t a[n_, k_] := (-1)^k/k! Sum[(-1)^p Binomial[k, p] Product[FactorialPower[p + 4*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2n} ] // Flatten (* _Jean-François Alcover_, Sep 01 2016 *)
%Y Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2.
%Y Cf. A091544 (first column), A091550 (second column divided by 12).
%Y Cf. A091748 (row sums), A091750 (alternating row sums).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Feb 27 2004