A091543 Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m >= 0.

1, 2, 1, 4, 6, 1, 8, 72, 12, 1, 16, 1440, 360, 20, 1, 32, 43200, 20160, 1120, 30, 1, 64, 1814400, 1814400, 123200, 2700, 42, 1, 128, 101606400, 239500800, 22422400, 491400, 5544, 56, 1, 256, 7315660800, 43589145600, 6098892800, 150368400

Offset: 1

Author

Wolfdieter Lang, Feb 13 2004

Keywords

• nonn
• easy
• tabl

Links

• P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
• P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• Wolfdieter Lang, First 10 rows.
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Crossrefs

Cf. A091547 (row sums), A091548 (alternating row sums).

For m = 0, 1, ..., 6 the column sequences are (without leading zeros): A000079 (powers of 2), A010796, A002674, A091535, A091544-6.

Formula

a(n, m) = m^(2*(n-m))*Pochhammer(1/m, n-m)*Pochhammer(2/m, n-m)/2 if n-1 >= m >= 1; a(n, 0) = 2^(n-1); otherwise 0.

E.g.f. for m = 1, 2, ... column (without leading zeros and offset n=1): (hypergeom([1/m, 2/m], [], (m^2)*x)-1)/2.

G.f. for m=1 column: x/(1-2*x); e.g.f.: (exp(2*x)-1)/2.

a(n, m) = (1/2)*Product_{j=0..n-m-1} (m*j+2)*(m*j+1), n >= m+1 >= 1, otherwise 0. From eq. 12 of the Blasiak et al. reference with r=m+2, s=2, k=2.