%I #18 Feb 28 2020 10:22:21

%S 1,56,16,1,10192,4928,776,48,1,3872960,2477440,575680,63360,3536,96,1,

%T 2517424000,1940556800,572868800,86163840,7326880,364800,10480,160,1,

%U 2497284608000,2210343116800,773352966400,143430604800,15836206400,1099612800,49056960,1398400,24520,240,1

%N Generalized Stirling2 array (8,2).

%C The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

%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 Wolfdieter Lang, <a href="/A092077/a092077.txt">First 6 rows</a>.

%H M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.

%F a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+6*(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=8, s=2.

%F Recursion: a(n, k) = sum(binomial(2, p)*fallfac(6*(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=8, 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 + 6(j-1), 2], {j, 1, n}], {p, 2, k}];

%t Table[a[n, k], {n, 1, 6}, {k, 2, 2n}] // Flatten (* _Jean-François Alcover_, Feb 28 2020 *)

%Y The generalized (k, 2)-Stirling2 arrays are, for k=2, ..., 7: A078739, A078740, A090438, A091534, A091746 and A091747.

%Y Cf. A091546, A091552 (first, resp. second column). A091757 (row sums). A091758 (alternating row sums).

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Feb 27 2004