A090216 Generalized Stirling2 array S_{5,5}(n,k).
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
Offset: 1
Examples
Triangle begins: [1]; [120,600,600,200,25,1]; [14400,504000,2664000,4608000,3501000,1350360,284800,33800,2225,75,1]; ...
Links
- P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004; Phys. Lett. A 309 (3-4) (2003) 198-205.
- Wolfdieter Lang, First 5 rows.
- M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.
Programs
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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-François Alcover, Mar 05 2014 *) -
Python
from sympy import binomial, factorial, ff def a(n, k): return sum((-1)**p * binomial(k, p) * ff(p, 5)**n for p in range(5, k+1)) * (-1)**k / factorial(k) # David Radcliffe, Jul 01 2025
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.
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