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A090216 Generalized Stirling2 array S_{5,5}(n,k). 7

%I #18 Aug 28 2019 17:07:31

%S 1,120,600,600,200,25,1,14400,504000,2664000,4608000,3501000,1350360,

%T 284800,33800,2225,75,1,1728000,371520000,7629120000,42762240000,

%U 97388280000,110386900800,70137648000,26920728000,6548346000,1039382000

%N Generalized Stirling2 array S_{5,5}(n,k).

%C The row length sequence for this array is [1, 6, 11, 16, 21, 26, 31,...]= A016861(n-1), n>=1.

%C The g.f. for the k-th column, (with leading zeros and k>=5) is G(k,x)= x^ceiling(k/5)*P(k,x)/product(1-fallfac(p,5)*x,p=5..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A090222(k,m)*x^m,m=0..kmax(k)), k>=5, with kmax(k) := floor(4*(k-5)/5)= A090223(k-5). For the recurrence of the G(k,x) see A090222.

%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 P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem.</a>

%H W. Lang, <a href="/A090216/a090216.txt">First 5 rows</a>.

%F a(n, k)= (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*fallfac(p, 5)^n, p=5..k), with fallfac(p, 5) := A008279(p, 5)=product(p+1-q, q=1..5); 5<= k <= 5*n, n>=1, else 0. From eq.(19) with r=5 of the Blasiak et al. reference.

%F E^n = sum_{k=5}^(5n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^5d^5/dx^5.

%e [1]; [120,600,600,200,25,1]; [14400,504000,2664000,4608000,3501000,1350360,284800,33800,2225,75,1]; ...

%t fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; Table[a[n, k], {n, 1, 5}, {k, 5, 5*n}] // Flatten (* _Jean-François Alcover_, Mar 05 2014 *)

%Y Cf. A090217, A090209 (row sums), A090218 (alternating row sums).

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Dec 01 2003

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