A091544 First column sequence of array A091746 ((6,2)-Stirling2).

1, 30, 2700, 491400, 150368400, 69470200800, 45155630520000, 39285398552400000, 44078217175792800000, 61973973349164676800000, 106719182107261573449600000, 220908706962031457040672000000, 541226332056977069749646400000000, 1548989762347068373623487996800000000

Offset: 1

Author

Wolfdieter Lang, Feb 13 2004

Keywords

• nonn
• changed

Comments

Also fifth column (m=4) sequence of triangle A091543.

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. A091535 (third column of A091543, first column of array A091534), A000407, A007696, A091746.

Programs

Mathematica

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

Formula

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

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

a(n) = fac4(4*n-3)*fac4(4*n-2)/2, n>=1, with fac4(4*n-3) = A007696(n) and fac4(4*n-2)/2 = A000407(n+1) (quartic- or 4-factorials).

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

a(n) = A091746(n, 2), n>=1.

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