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

1,2

Comments

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

Links

Table of n, a(n) for n=1..36.

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.

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).

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 *)

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).

Sequence in context: A107676 A005932 A109737 * A333076 A033376 A300449

Adjacent sequences: A092074 A092075 A092076 * A092078 A092079 A092080

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Feb 27 2004

Status

approved