0,4

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.

W. Lang, First 8 rows.

a(n, m)= Bell(m;n-(m-1)), n>= m-1 >=0, with Bell(m;k) := sum(S2(m;k, p), p=m..m*k), where S2(m;k, p) := (((-1)^p)/p!)*sum(((-1)^r)*binomial(p, r)*fallfac(p, r)^k, r=m..p); with fallfac(n, m) := A008279(n, m) (falling factorials) and m<=p<=k*m, k>=1, m=1, 2, ..., else 0. From eqs.(6) with r=s->m and eq.(19) with S_{r, r}(n, k)-> S2(r;n, k) of the Blasiak et al. reference.

a(n, m)= (sum(fallfac(k, m)^(n-(m-1)), k=m..infinity))/exp(1), n>= m-1 >=0, else 0. From eq.(26) with r->m of the Schork reference which is rewritten eq.(11) of the original Blasiak et al. reference.

E.g.f. m-th column (no leading zeros): (sum((exp(fallfac(k, m)*x))/k!, k=m..infinity) + A000522(m)/m!)/exp(1). Rewritten from the top of p. 4656 of the Schork reference.

[1]; [1, 1]; [2, 1, 1]; [5, 7, 1, 1]; [15, 87, 34, 1, 1]; ...

Sequence in context: A064814 A051012 A064644 this_sequence A024462 A049252 A098315

Adjacent sequences: A090207 A090208 A090209 this_sequence A090211 A090212 A090213

nonn,easy,tabl

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