A091549 Second column (k=3) sequence of array A078740 ((3,2)-Stirling2) divided by 6.

1, 28, 960, 43200, 2520000, 186278400, 17069875200, 1902071808000, 253487646720000, 39833773056000000, 7291173820170240000, 1538106259064094720000, 370502654756909875200000, 101080724272141565952000000, 31008222182732149555200000000, 10627137906465962295558144000000

Offset: 2

Author

Wolfdieter Lang, Feb 13 2004

Keywords

• nonn
• easy
• changed

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.

Crossrefs

Cf. A078740.

Programs

Maple

A091549 := proc(n)
    n!*(n+1)!*(-3 + (n+2)*(n+1)/2)/(3!)^2 ;
end proc:
seq(A091549(n),n=2..30) ; # R. J. Mathar, Jul 27 2022

Mathematica

a[n_] := n!*(n+1)!*((n+2)*(n+1)/2 - 3) / 36; Array[a, 16, 2] (* Amiram Eldar, Sep 01 2025 *)

Formula

a(n) = n!*(n+1)!*(-3 + (n+2)*(n+1)/2)/(3!)^2, n>=2.

E.g.f.: (hypergeom([2, 3], [], x) - 3*hypergeom([1, 2], [], x) + 2)/(3!)^2.

a(n) = Product_{j=0..n-1} (j+2) * (-3 * Product_{j=0..n-1} (j+1) + Product_{j=0..n-1} (j+3))/(3!)^2, n>=2. From eq.12 of the Blasiak et al. reference with r=3, s=2, k=3.

D-finite with recurrence a(n) + (-n^2-7*n-24)*a(n-1) + 12*(n^2+4*n+6)*a(n-2) - 36*n*(n+1)*a(n-3) = 0. - R. J. Mathar, Jul 27 2022

a(n) ~ Pi * n^(2*n+4) / (36 * exp(2*n)). - Amiram Eldar, Sep 01 2025