A091550 Second column (k=3) sequence of array A091746 ((6,2)-Stirling2) divided by 12.

1, 160, 39900, 15120000, 8202070800, 6058891238400, 5860547004312000, 7196668193594880000, 10944624305020966560000, 20199809308312018344960000, 44490168120726255724917120000, 115290834599202214240544256000000, 347284815748143369922163257920000000

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. A091539 (second column of (5, 2)-Stirling2 array), A091550 (second column of (7, 2)-Stirling2 array), A091746.

Programs

Mathematica

a[n_] := 2^(4*n) * Pochhammer[1/2, n] * (-3 * Pochhammer[1/4, n] + Pochhammer[3/4, n])/(3!*12); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = (2^(4*n)) * risefac(1/2, n) * (-3*risefac(1/4, n) + risefac(3/4, n))/(3!*12), n>=2, with risefac(x, n) = Pochhammer(x, n).

E.g.f.: (hypergeom([1/2, 3/4], [], 16*x) - 3*hypergeom([1/4, 1/2], [], 16*x) + 2)/(3!*12).

a(n) = (2^n) * Product_{j=0..n-1} (2*j+1) * (-3*Product_{j=0..n-1} (4*j+1) + Product_{j=0..n-1} (4*j+3))/(3!*12), n>=2. From eq.12 of the Blasiak et al. reference with r=6, s=2, k=3.

a(n) ~ sqrt(Pi) * 2^(4*n-2) * n^(2*n+1/4) / (9 * Gamma(3/4) * exp(2*n)). - Amiram Eldar, Aug 30 2025