1,2

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.

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

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

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.

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.

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

Sequence in context: A129764 A027552 A098210 this_sequence A027492 A001728 A059383

Adjacent sequences: A090210 A090211 A090212 this_sequence A090214 A090215 A090216

sign,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003