A091552 -id:A091552 - cp's OEIS frontend

A092077 Generalized Stirling2 array (8,2).

1, 56, 16, 1, 10192, 4928, 776, 48, 1, 3872960, 2477440, 575680, 63360, 3536, 96, 1, 2517424000, 1940556800, 572868800, 86163840, 7326880, 364800, 10480, 160, 1, 2497284608000, 2210343116800, 773352966400, 143430604800, 15836206400, 1099612800, 49056960, 1398400, 24520, 240, 1

Offset: 1

Wolfdieter Lang, Feb 27 2004

The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Wolfdieter Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

The generalized (k, 2)-Stirling2 arrays are, for k=2, ..., 7: A078739, A078740, A090438, A091534, A091746 and A091747.

Cf. A091546, A091552 (first, resp. second column). A091757 (row sums). A091758 (alternating row sums).

a[n_, k_] := ((-1)^k/k!) Sum[(-1)^p Binomial[k, p] Product[FactorialPower[ p + 6(j-1), 2], {j, 1, n}], {p, 2, k}];
Table[a[n, k], {n, 1, 6}, {k, 2, 2n}] // Flatten (* Jean-François Alcover, Feb 28 2020 *)

a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+6*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=8, s=2.

Recursion: a(n, k) = sum(binomial(2, p)*fallfac(6*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=8, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

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

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Feb 13 2004

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.

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

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 *)

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

Offset corrected by Amiram Eldar, Aug 30 2025

cp's OEIS Frontend

A092077 Generalized Stirling2 array (8,2).

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