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A091543 Triangle built from first column sequences of generalized Stirling2 arrays (m+2,2)-Stirling2, m>=0. 6
1, 2, 1, 4, 6, 1, 8, 72, 12, 1, 16, 1440, 360, 20, 1, 32, 43200, 20160, 1120, 30, 1, 64, 1814400, 1814400, 123200, 2700, 42, 1, 128, 101606400, 239500800, 22422400, 491400, 5544, 56, 1, 256, 7315660800, 43589145600, 6098892800, 150368400 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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..41.

W. Lang, First 10 rows.

FORMULA

a(n, m)= m^(2*(n-m))*pochhammer(1/m, n-m)*pochhammer(2/m, n-m)/2 if n-1>= m>=1; a(n, 0)= 2^(n-1); else 0.

E.g.f. for m=1, 2, ... column (without leading zeros and offset n=1): (hypergeom([1/m, 2/m], [], (m^2)*x)-1)/2.

G.f. for m=1 column: x/(1-2*x); e.g.f.: (exp(2*x)-1)/2.

a(n, m)= product((m*j+2)*(m*j+1), j=0..n-m-1)/2, n>=m+1>=1, else 0. From eq.12 of the Blasiak et al. reference with r=m+2, s=2, k=2.

CROSSREFS

Cf. A091547 (row sums), A091548 (alternating row sums).

For m=0, 1, ..., 6 the column sequences are (without leading zeros): A000079 (powers of 2), A010796, A002674, A091535, A091544-6.

Sequence in context: A185947 A268472 A079474 * A330858 A059575 A338805

Adjacent sequences:  A091540 A091541 A091542 * A091544 A091545 A091546

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Feb 13 2004

STATUS

approved

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Last modified June 20 02:34 EDT 2021. Contains 345154 sequences. (Running on oeis4.)