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%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
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