A091747 Generalized Stirling2 array (7,2).

1, 42, 14, 1, 5544, 3192, 588, 42, 1, 1507968, 1165248, 321552, 41496, 2688, 84, 1, 696681216, 655966080, 232606080, 41497344, 4143552, 240240, 7980, 140, 1, 489070213632, 533531142144, 226306918656, 50249808000, 6575950080

Offset

1,2

Comments

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

References

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Links

Table of n, a(n) for n=1..30.

W. Lang, First 6 rows.

Formula

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+5*(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=7, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(5*(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=7, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

Crossrefs

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2, A091746 (6, 2)-Stirling2.

Cf. A091545 (first column).

Cf. A091749 (row sums), A091751 (alternating row sums).

Sequence in context: A176920 A037938 A371699 * A030434 A194710 A334148

Adjacent sequences: A091744 A091745 A091746 * A091748 A091749 A091750

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Feb 27 2004

Status

approved