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A091534
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Generalized Stirling2 array (5,2).
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11
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1, 20, 10, 1, 1120, 1040, 290, 30, 1, 123200, 161920, 71320, 14040, 1340, 60, 1, 22422400, 37452800, 22097600, 6263040, 958720, 82800, 4000, 100, 1, 6098892800, 12222918400, 8928102400, 3257116800, 675281600, 84782880, 6625920, 322000
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OFFSET
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1,2
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COMMENTS
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The row length sequences for this array is [1,3,5,7,9,11,...]=A005408(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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Table of n, a(n) for n=1..33.
W. Lang, First 6 rows.
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FORMULA
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a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+3*(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=5, s=2.
Recursion: a(n, k)=sum(binomial(2, p)*fallfac(3*(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=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
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CROSSREFS
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Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2.
Cf. A072019 (row sums), A091537 (alternating row sums).
Sequence in context: A078080 A216289 A136010 * A033966 A033340 A040383
Adjacent sequences: A091531 A091532 A091533 * A091535 A091536 A091537
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Wolfdieter Lang, Jan 23 2004
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STATUS
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approved
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