A091746 Generalized Stirling2 array (6,2).

1, 30, 12, 1, 2700, 1920, 426, 36, 1, 491400, 478800, 162540, 25344, 1956, 72, 1, 150368400, 181440000, 80451000, 17624880, 2130660, 147840, 5820, 120, 1, 69470200800, 98424849600, 52905560400, 14618016000, 2346624000, 232202880

Offset

1,2

Comments

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

References

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Links

Table of n, a(n) for n=1..31.

W. Lang, First 6 rows.

Formula

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+4*(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=6, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(4*(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=6, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

Mathematica

a[n_, k_] := (-1)^k/k! Sum[(-1)^p Binomial[k, p] Product[FactorialPower[p + 4*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2n} ] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

Crossrefs

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2.

Cf. A091544 (first column), A091550 (second column divided by 12).

Cf. A091748 (row sums), A091750 (alternating row sums).

Sequence in context: A350085 A040875 A131773 * A040874 A147454 A147077

Adjacent sequences: A091743 A091744 A091745 * A091747 A091748 A091749

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Feb 27 2004

Status

approved