A090211 Alternating row sums of array A078739 ((2,2)-Stirling2).

1, -1, -1, 41, -375, -3001, 177063, -990543, -144800527, 3644593711, 214013895023, -12488200175463, -553322483517383, 61495192102867639, 2469939623420627543, -448608666325921194271, -19104207797417792353951, 4742067751530355028847327

Offset: 1

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• sign
• easy

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.

Crossrefs

Cf. -A000587(n) from Stirling2 case A008277 with a(0) := -1. A020556 (non-alternating sum, generalized Bell-numbers).

Programs

Mathematica

a[n_] := Sum[(-1)^k FactorialPower[k, 2]^n/k!, {k, 2, Infinity}]*E; Array[a, 18] (* Jean-François Alcover, Sep 01 2016 *)

Formula

a(n) := sum( A078739(n, m)*(-1)^m, m=2..2*n), n>=1. a(0) := +1 may be added.

a(n) = sum(((-1)^k)*(fallfac(k, 2)^n)/k!, k=2..infinity)*exp(1), with fallfac(k, 2)=A008279(k, 2)=k*(k-1) and n>=1. This produces also a(0)=1.

E.g.f. if a(0)=1 is added: exp(1)*(sum(((-1)^k)*exp(k*(k-1)*x)/k!, k=2..infinity)). Similar to derivation on top p. 4656 of the Schork reference.