A090213 - OEIS

%I #7 Sep 01 2016 10:55:41

%S 1,-15,1169,-154079,-148969375,778633335441,-4003896394897551,

%T 27901641934428560705,-268555885416357907647039,

%U 3460225909437698652973995569,-56404253763542830420650221273263,1050004356721541004548911018674177377

%N Alternating row sums of array A090214 ((4,4)-Stirling2).

%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>

%F a(n) := sum( A090214(n, k)*(-1)^k, k=4..4*n), n>=1. a(0) := 1 may be added.

%F a(n) = sum(((-1)^k)*(fallfac(k, 4)^n)/k!, k=4..infinity)*exp(1), with fallfac(k, 4)=A008279(k, 4)=k*(k-1)*(k-2)*(k-3) and n>=1. This produces also a(0)=1.

%F E.g.f. if a(0)=1 is added: exp(1)*(sum(((-1)^k)*exp(fallfac(k, 4)*x)/k!, k=4..infinity) + A000166(3)/3!). with the subfactorials A000166. A000166(3)/3!=1/3. Similar to derivation on top of p. 4656 of the Schork reference.

%t a[n_] := Sum[(-1)^k FactorialPower[k, 4]^n/k!, {k, 2, Infinity}]*E; Array[a, 12] (* _Jean-François Alcover_, Sep 01 2016 *)

%Y Cf. A000587, A090211-2. A071379 (non-alternating sum, generalized Bell-numbers).

%K sign,easy

%O 1,2

%A _Wolfdieter Lang_, Dec 01 2003