A091551 Second column (k=3) sequence of array ((7,2)-Stirling2) divided by 14.

1, 228, 83232, 46854720, 38109367296, 42479241412608, 62290218157719552, 116373513947009679360, 270010358636135897235456, 762020881523854021734432768, 2571195906705444158241905836032, 10223478528521152233103572672184320, 47315411140234001777600560898513043456

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. A091550 (second column of (6, 2)-Stirling2 array), A091552 (second column of (8, 2)-Stirling2 array).

Programs

Mathematica

a[n_] := 5^(2*n) * Pochhammer[2/5, n] * (-3 * Pochhammer[1/5, n] + Pochhammer[3/5, n])/(3!*14); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

Formula

a(n) = Product_{j=0..n-1} (5*j+2) * (-3*Product_{j=0..n-1} (5*j+1) + Product_{j=0..n-1} (5*j+3)/(3!*14), n>=2. From eq.12 of the Blasiak et al. reference with r=7, s=2, k=3.

a(n) = (5^(2*n))*risefac(2/5, n) * (-3*risefac(1/5, n) + risefac(3/5, n))/(3!*14), n>=2, with risefac(x, n) = Pochhammer(x, n).

E.g.f.: (hypergeom([2/5, 3/5], [], 25*x) - 3*hypergeom([1/5, 2/5], [], 25*x) + 2)/(3!*14).

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

Extensions

Offset corrected by Amiram Eldar, Aug 30 2025