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A090214
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Generalized Stirling2 array S_{4,4}(n,k).
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6
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1, 24, 96, 72, 16, 1, 576, 13824, 50688, 59904, 30024, 7200, 856, 48, 1, 13824, 1714176, 21606912, 76317696, 110160576, 78451200, 30645504, 6976512, 953424, 78400, 3760, 96, 1, 331776, 207028224, 8190885888, 74684104704, 253100173824
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The row length sequence for this array is [1,5,9,13,17,...]= A016813(n-1), n>=1.
The g.f. for the k-th column, (with leading zeros and k>=4) is G(k,x)= x^ceiling(k/4)*P(k,x)/product(1-fallfac(p,4)*x,p=4..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A090221(k,m)*x^m,m=0..kmax(k)), k>=4, with kmax(k) := A057353(k-4)= floor(3*(k-4)/4). For the recurrence of the G(k,x) see A090221.
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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
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
W. Lang, First 4 rows.
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FORMULA
| a(n, k)= (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*fallfac(p, 4)^n, p=4..k), with fallfac(p, 4) := A008279(p, 4)=p*(p-1)*(p-2)*(p-3); 4<= k <= 4*n, n>=1, else 0. From eq.(19) with r=4 of the Blasiak et al. reference.
E^n = sum_{k=4}^(4n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^4d^4/dx^4.
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EXAMPLE
| [1]; [24,96,72,16,1]; [576,13824,50688,59904,30024,7200,856,48,1]; ...
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CROSSREFS
| Cf. A090215, A071379 (row sums), A090213 (alternating row sums).
Sequence in context: A010012 A076799 A055671 * A103251 A198387 A057102
Adjacent sequences: A090211 A090212 A090213 * A090215 A090216 A090217
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003
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