A090440 Generalized Stirling2 array (4,3).

1, 24, 36, 12, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 172800, 1339200, 2808000, 2420640, 1025280, 232920, 29400, 2040, 72, 1, 36288000, 471744000, 1643846400, 2381702400, 1745755200, 721224000, 178941600, 27624960, 2689920, 163800, 6000

Offset

1,2

Comments

The row lengths for this array are [1,4,7,10,13,16,...] = A016777(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..33.

W. Lang, First 6 rows.

Formula

Recursion: a(n, k)=sum(binomial(3, p)*fallfac(n-1-p+k, 3-p)*a(n-1, k-p), p=0..3), n>=2, 3<=k<=3*n, a(1, 3)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=3. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+j-1, 3), j=1..n), p=3..k), n>=1, 3<=k<=3*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=3.

Mathematica

ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, _] = 0; Flatten[Table[a[n, k], {n, 1, 5}, {k, 3, 3 n}]] (* Jean-François Alcover, Sep 01 2011, after given recursion *)

Crossrefs

Cf. A090438 (4, 2)-Stirling2.

Cf. A070531 (row sums), A091028 (alternating row sums).

Sequence in context: A179188 A102505 A112064 * A091192 A067807 A224907

Adjacent sequences: A090437 A090438 A090439 * A090441 A090442 A090443

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Dec 23 2003

Status

approved