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

Author

Wolfdieter Lang, Dec 23 2003

Keywords

• nonn
• easy
• tabf

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

• W. Lang, First 6 rows.

Crossrefs

Cf. A090438 (4, 2)-Stirling2.

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

Programs

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 *)

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.