A091746 Generalized Stirling2 array (6,2).
1, 30, 12, 1, 2700, 1920, 426, 36, 1, 491400, 478800, 162540, 25344, 1956, 72, 1, 150368400, 181440000, 80451000, 17624880, 2130660, 147840, 5820, 120, 1, 69470200800, 98424849600, 52905560400, 14618016000, 2346624000, 232202880
Offset: 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
- W. Lang, First 6 rows.
Crossrefs
Programs
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Mathematica
a[n_, k_] := (-1)^k/k! Sum[(-1)^p Binomial[k, p] Product[FactorialPower[p + 4*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2n} ] // Flatten (* Jean-François Alcover, Sep 01 2016 *)
Formula
a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+4*(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=6, s=2.
Recursion: a(n, k)=sum(binomial(2, p)*fallfac(4*(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=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
Comments