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A091747
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Generalized Stirling2 array (7,2).
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12
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1, 42, 14, 1, 5544, 3192, 588, 42, 1, 1507968, 1165248, 321552, 41496, 2688, 84, 1, 696681216, 655966080, 232606080, 41497344, 4143552, 240240, 7980, 140, 1, 489070213632, 533531142144, 226306918656, 50249808000, 6575950080
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The sequence of row lengths for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.
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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.
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LINKS
| W. Lang, First 6 rows.
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FORMULA
| a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+5*(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=7, s=2.
Recursion: a(n, k)=sum(binomial(2, p)*fallfac(5*(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=7, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
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CROSSREFS
| Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2, A091534 (5, 2)-Stirling2, A091746 (6, 2)-Stirling2.
Cf. A091545 (first column).
Cf. A091749 (row sums), A091751 (alternating row sums).
Sequence in context: A203484 A176920 A037938 * A030434 A033362 A033976
Adjacent sequences: A091744 A091745 A091746 * A091748 A091749 A091750
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 27 2004
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