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A091534 Generalized Stirling2 array (5,2). 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The row length sequences for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

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.

LINKS

Table of n, a(n) for n=1..33.

W. Lang, First 6 rows.

FORMULA

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).

CROSSREFS

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

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Jan 23 2004

STATUS

approved

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Last modified October 1 06:15 EDT 2014. Contains 247503 sequences.