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Search: id:A078740
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| A078740 |
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Triangle of generalized Stirling numbers S_{3,2}(n,k) read by rows (n>=1, 2<=k<=2n). |
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17
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| 1, 6, 6, 1, 72, 168, 96, 18, 1, 1440, 5760, 6120, 2520, 456, 36, 1, 43200, 259200, 424800, 285120, 92520, 15600, 1380, 60, 1, 1814400, 15120000, 34776000, 33566400, 16304400, 4379760, 682200, 62400, 3270, 90, 1, 101606400, 1117670400
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The sequence of row lengths of this array is [1,3,5,7,...]=A005408(n-1), n>=1.
For the scaled array s2_{3,2}(n,k) := a(n,k)*k!/((n+1)!*n!) see A090452.
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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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P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
W. Lang, First 6 rows.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
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FORMULA
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Recursion: a(n, k)=sum(binomial(2, p)*fallfac(n-1-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 1)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=3, s=2. 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)*(3-2), 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=3, s=2.
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EXAMPLE
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1; 6,6,1; ...
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CROSSREFS
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Row sums give A078738. Cf. A078739.
Sequence in context: A098267 A122193 A098369 this_sequence A021155 A003676 A033259
Adjacent sequences: A078737 A078738 A078739 this_sequence A078741 A078742 A078743
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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njas, Dec 21 2002
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EXTENSIONS
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Edited by Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003
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