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A091039 Triangle of scaled second column sequences of (k,k)-Stirling2 arrays. 3


%S 1,3,1,7,8,1,15,52,30,1,31,320,756,144,1,63,1936,18360,17856,840,1,

%T 127,11648,441936,2156544,619200,5760,1,255,69952,10614240,259117056,

%U 447552000,29548800,45360,1,511,419840,254788416,31102009344

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

%C 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.

%D P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

%D M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

%H W. Lang, <a href="/A091039/a091039.txt">First 9 rows</a>.

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

%F 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).

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

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

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Jan 23 2004

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