A091552 Second column (k=3) sequence of array A092077 ((8,2)-Stirling2) divided by 16.

1, 308, 154840, 121284800, 138146444800, 216595133081600, 448169865375232000, 1184352885735219200000, 3894384547720687820800000, 15599967808704696966348800000, 74806554280938737689393561600000, 423166219648146647212581257216000000, 2788777788711380327670376970321920000000

Offset

2,2

Links

Table of n, a(n) for n=2..14.

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.

Formula

E.g.f.: (hypergeom([1/3, 1/2], [], 36*x) - 3*hypergeom([1/6, 1/3], [], 36*x) + 2)/(3!*16).

a(n) = (2^n) * Product_{j=0..n-1} (3*j+1) * (-3*Product_{j=0..n-1} (6*j+1) + Product_{j=0..n-1} (6*j+3))/(3!*16), n>=2. From eq.12 of the Blasiak et al. reference with r=8, s=2, k=3.

a(n) = (2^(2*n-5)) * (3^(2*n-1)) * risefac(1/3, n) * (-3*risefac(1/6, n) + risefac(1/2, n)), n>=2, with risefac(x, n) = Pochhammer(x, n).

a(n) ~ sqrt(Pi) * 2^(2*n-4) * 3^(2*n-1) * n^(2*n-1/6) / (Gamma(1/3) * exp(2*n)). - Amiram Eldar, Aug 30 2025

Mathematica

a[n_] := 2^(2*n-5) * 3^(2*n-1) * Pochhammer[1/3, n] * (-3 * Pochhammer[1/6, n] + Pochhammer[1/2, n]); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

Crossrefs

Cf. A091551 (second column of (7, 2)-Stirling2 array).

Sequence in context: A053172 A343273 A337955 * A364936 A159004 A031782

Adjacent sequences: A091549 A091550 A091551 * A091553 A091554 A091555

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 13 2004

Status

approved