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A090216 Generalized Stirling2 array S_{5,5}(n,k). 7
1, 120, 600, 600, 200, 25, 1, 14400, 504000, 2664000, 4608000, 3501000, 1350360, 284800, 33800, 2225, 75, 1, 1728000, 371520000, 7629120000, 42762240000, 97388280000, 110386900800, 70137648000, 26920728000, 6548346000, 1039382000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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.

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..28.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

W. Lang, First 5 rows.

FORMULA

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.

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.

EXAMPLE

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

MATHEMATICA

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-Fran├žois Alcover, Mar 05 2014 *)

CROSSREFS

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

Sequence in context: A152622 A100145 A234437 * A265041 A280425 A235768

Adjacent sequences:  A090213 A090214 A090215 * A090217 A090218 A090219

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Dec 01 2003

STATUS

approved

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Last modified September 27 00:14 EDT 2020. Contains 337378 sequences. (Running on oeis4.)