A092077 Generalized Stirling2 array (8,2).

1, 56, 16, 1, 10192, 4928, 776, 48, 1, 3872960, 2477440, 575680, 63360, 3536, 96, 1, 2517424000, 1940556800, 572868800, 86163840, 7326880, 364800, 10480, 160, 1, 2497284608000, 2210343116800, 773352966400, 143430604800, 15836206400, 1099612800, 49056960, 1398400, 24520, 240, 1

Offset: 1

Author

Wolfdieter Lang, Feb 27 2004

Keywords

• nonn
• easy
• tabf

Comments

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

Links

• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• Wolfdieter Lang, First 6 rows.
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Crossrefs

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

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

Programs

Mathematica

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}];
Table[a[n, k], {n, 1, 6}, {k, 2, 2n}] // Flatten (* Jean-François Alcover, Feb 28 2020 *)

Formula

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.

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).