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Search: id:A090440
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| A090440 |
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Generalized Stirling2 array (4,3). |
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+0
3
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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The row lengths for this array are [1,4,7,10,13,16,...] = A016777(n-1), n>=1.
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REFERENCES
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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.
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LINKS
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W. Lang, First 6 rows.
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FORMULA
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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.
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CROSSREFS
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Cf. A090438 (4, 2)-Stirling2.
Cf. A070531 (row sums), A091028 (alternating row sums).
Sequence in context: A025102 A102505 A112064 this_sequence A091192 A067807 A067341
Adjacent sequences: A090437 A090438 A090439 this_sequence A090441 A090442 A090443
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003
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