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%I #17 Aug 29 2019 17:25:27
%S 1,24,36,12,1,1440,5760,6120,2520,456,36,1,172800,1339200,2808000,
%T 2420640,1025280,232920,29400,2040,72,1,36288000,471744000,1643846400,
%U 2381702400,1745755200,721224000,178941600,27624960,2689920,163800,6000
%N Generalized Stirling2 array (4,3).
%C The row lengths for this array are [1,4,7,10,13,16,...] = A016777(n-1), n>=1.
%D P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
%D M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
%H W. Lang, <a href="/A090440/a090440.txt">First 6 rows</a>.
%F Recursion: a(n, k)=sum(binomial(3, p)*fallfac(n-1-p+k, 3-p)*a(n-1, k-p), p=0..3), n>=2, 3<=k<=3*n, a(1, 3)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=3. 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), j=1..n), p=3..k), n>=1, 3<=k<=3*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=3.
%t ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, _] = 0; Flatten[Table[a[n, k], {n, 1, 5}, {k, 3, 3 n}]] (* _Jean-François Alcover_, Sep 01 2011, after given recursion *)
%Y Cf. A090438 (4, 2)-Stirling2.
%Y Cf. A070531 (row sums), A091028 (alternating row sums).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Dec 23 2003
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