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A092077 Generalized Stirling2 array (8,2). 5
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 (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.

LINKS

Table of n, a(n) for n=1..36.

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.

FORMULA

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).

MATHEMATICA

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-Fran├žois Alcover, Feb 28 2020 *)

CROSSREFS

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).

Sequence in context: A107676 A005932 A109737 * A333076 A033376 A300449

Adjacent sequences:  A092074 A092075 A092076 * A092078 A092079 A092080

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Feb 27 2004

STATUS

approved

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Last modified May 27 05:48 EDT 2020. Contains 334649 sequences. (Running on oeis4.)