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%I #37 Nov 20 2019 21:52:27
%S 1,6,6,1,72,168,96,18,1,1440,5760,6120,2520,456,36,1,43200,259200,
%T 424800,285120,92520,15600,1380,60,1,1814400,15120000,34776000,
%U 33566400,16304400,4379760,682200,62400,3270,90,1,101606400,1117670400
%N Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n).
%C The sequence of row lengths of this array is [1,3,5,7,...] = A005408(n-1), n>=1.
%C For the scaled array s2_{3,2}(n,k) := a(n,k)*k!/((n+1)!*n!) see A090452.
%H Michael De Vlieger, <a href="/A078740/b078740.txt">Table of n, a(n) for n = 1..10000</a> (rows 1 <= n <= 100, flattened).
%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.
%H A. Dzhumadildaev and D. Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, 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]
%H Askar Dzhumadil'daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%H W. Lang, <a href="/A078740/a078740.txt">First 6 rows</a>.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
%H M. Schork, <a href="http://dx.doi.org/10.1088/0305-4470/36/16/314">On the combinatorics of normal ordering bosonic operators and deforming it</a>, J. Phys. A 36 (2003) 4651-4665.
%F 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).
%F 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.
%F 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
%e 1;
%e 6, 6, 1;
%e 72, 168, 96, 18, 1;
%e ...
%t 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 *)
%Y Row sums give A078738. Cf. A078739.
%Y Cf. A005408, A090452.
%K nonn,tabf,easy
%O 1,2
%A _N. J. A. Sloane_, Dec 21 2002
%E Edited by _Wolfdieter Lang_, Dec 23 2003
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