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 A090440 Generalized Stirling2 array (4,3). 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified August 9 12:53 EDT 2022. Contains 356026 sequences. (Running on oeis4.)