A078740 Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).

1, 6, 6, 1, 72, 168, 96, 18, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 43200, 259200, 424800, 285120, 92520, 15600, 1380, 60, 1, 1814400, 15120000, 34776000, 33566400, 16304400, 4379760, 682200, 62400, 3270, 90, 1, 101606400, 1117670400

Offset: 1

Author

N. J. A. Sloane, Dec 21 2002

Keywords

• nonn
• tabf
• easy

Comments

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

For the scaled array s2_{3,2}(n,k) := a(n,k)*k!/((n+1)!*n!) see A090452.

Examples

			1;
6, 6, 1;
72, 168, 96, 18, 1;
...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100, flattened).
• Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
• P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
• A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
• Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
• W. Lang, First 6 rows.
• Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Crossrefs

Row sums give A078738. Cf. A078739.

Cf. A005408, A090452.

Programs

Mathematica

a[n_, k_] := (-1)^k*n!*(n+1)!*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!); Table[a[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

Formula

Recursion: a(n, k) = Sum(binomial(2, p)*fallfac(n-1-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 1)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

a(n, k) = (((-1)^k)/k!)*Sum(((-1)^p)*binomial(k, p)*product(fallfac(p+(j-1)*(3-2), 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=3, s=2.

a(n, k) = (-1)^k n! (n+1)! 3F2(2-k, n+1, n+2; 2, 3; 1) / (2(k-2)!). - Jean-François Alcover, Dec 04 2013

Extensions

Edited by Wolfdieter Lang, Dec 23 2003