1,2

The row length sequence for this array is [1,5,9,13,17,...]= A016813(n-1), n>=1.

The g.f. for the k-th column, (with leading zeros and k>=4) is G(k,x)= x^ceiling(k/4)*P(k,x)/product(1-fallfac(p,4)*x,p=4..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A090221(k,m)*x^m,m=0..kmax(k)), k>=4, with kmax(k) := A057353(k-4)= floor(3*(k-4)/4). For the recurrence of the G(k,x) see A090221.

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.

W. Lang, First 4 rows.

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

a(n, k)= (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*fallfac(p, 4)^n, p=4..k), with fallfac(p, 4) := A008279(p, 4)=p*(p-1)*(p-2)*(p-3); 4<= k <= 4*n, n>=1, else 0. From eq.(19) with r=4 of the Blasiak et al. reference.

[1]; [24,96,72,16,1]; [576,13824,50688,59904,30024,7200,856,48,1]; ...

Cf. A090215, A071379 (row sums), A090213 (alternating row sums).

Sequence in context: A010012 A076799 A055671 this_sequence A103251 A057102 A057103

Adjacent sequences: A090211 A090212 A090213 this_sequence A090215 A090216 A090217

nonn,easy,tabf

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