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A091746 Generalized Stirling2 array (6,2). 7
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A073401 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

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Last modified August 17 11:06 EDT 2019. Contains 326057 sequences. (Running on oeis4.)