A090212 Alternating row sums of array A078741 ((3,3)-Stirling2).
1, -4, 73, -3241, 223546, -10884061, -5437091357, 4560715140638, -2741631069546683, 1315509914960956853, -135771066929217673256, -969783690708328561039261, 1943740128890758048004419957, -2140191682145533094039398047820
Offset: 1
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
- P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
Programs
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Mathematica
a[n_] := -Sum[(-1)^k FactorialPower[k, 3]^n/k!, {k, 2, Infinity}]*E; Array[a, 14] (* Jean-François Alcover, Sep 01 2016 *)
Formula
a(n) := sum( A078741(n, k)*(-1)^(k+1), k=3..3*n), n>=1. a(0) := -1 may be added.
a(n) = -sum(((-1)^k)*(fallfac(k, 3)^n)/k!, k=3..infinity)*exp(1), with fallfac(k, 3)=A008279(k, 3)=k*(k-1)*(k-2) and n>=1. This produces also a(0)=-1.
E.g.f. if a(0)=-1 is added: -exp(1)*(sum(((-1)^k)*exp(fallfac(k, 3)*x)/k!, k=3..infinity)+1/2). Similar to derivation on top of p. 4656 of the Schork reference.