A091039 Triangle of scaled second column sequences of (k,k)-Stirling2 arrays.

1, 3, 1, 7, 8, 1, 15, 52, 30, 1, 31, 320, 756, 144, 1, 63, 1936, 18360, 17856, 840, 1, 127, 11648, 441936, 2156544, 619200, 5760, 1, 255, 69952, 10614240, 259117056, 447552000, 29548800, 45360, 1, 511, 419840, 254788416, 31102009344

Offset

1,2

Comments

a(n-1,k)= S2_{k,k}(n,k+1)/(k*k!), n>=2, 1<=k<= n-1, with S2_{k,k} the r=k,s=k Stirling2 array S_{k,k} of the Blasiak et al. reference.

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..40.

W. Lang, First 9 rows.

Formula

a(n, k)=(k!^(n-k+1))*((k+1)^(n-k+1)-1)/(k*k!) if n >= k >= 1, else 0.

G.f. column k (without leading zeros): 1/((1-(k+1)!*x)*(1-k!*x)) = (1/(1-(k+1)!*x) - 1/(1-k!*x))/(k*k!*x).

Example

[1];[3,1];[7,8,1];[15,52,30,1];...

Crossrefs

Cf. A091040 (row sums), A091041 (alternating row sums).

Sequence in context: A058606 A135284 A016647 * A217594 A340616 A120472

Adjacent sequences: A091036 A091037 A091038 * A091040 A091041 A091042

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Jan 23 2004

Status

approved