A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.
1, 104, 16192, 3745280, 1222291840, 537758144000, 307503360102400, 221965373351321600, 197530935371241472000, 212553938009841139712000, 272115940122123843665920000, 408828811133790954169303040000, 712427095375430807967713198080000, 1425431682224708301179257251430400000
Offset: 2
Links
- Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.
Programs
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Mathematica
a[n_] := 3^(2*n) * Pochhammer[2/3, n] * (n! - 3 * Pochhammer[1/3, n])/(3!*10); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)
Formula
a(n) = A091534(n, 3)/10, n >= 2.
a(n) = Product_{j=0..n-1} (3*j + 2)*(Product_{j=0..n-1} (3*(j+1)) - 3*Product_{j=0..n-1} (3*j + 1))/(3!*10). From eq. (12) of the Blasiak et al. reference for r=5, s=2 and k=3.
a(n) = (3^(2*n))*risefac(2/3, n)*(n!-3*risefac(1/3, n))/(3!*10), with risefac(x, n) = Pochhammer(x, n).
a(n) = (fac3(3*n-1)/10)*(fac3(3*n) - 3*fac3(3*n-2))/3!, with fac3(3*n) = A032031(n) = n!*3^n, fac3(3*n-1) = A008544(n) and fac3(3*n-2) = A007559(n) (triple factorials: fac3(n) = A007661(n)).
E.g.f.: (hypergeom([2/3, 1], [], 9*x)-3*hypergeom([1/3, 2/3], [], 9*x)+2)/(3!*10).
a(n) ~ Pi * 3^(2*n) * n^(2*n + 2/3) / (30 * Gamma(2/3) * exp(2*n)). - Amiram Eldar, Aug 30 2025