1,2

The row length sequence for this array is [1, 6, 11, 16, 21, 26, 31,...]= A016861(n-1), n>=1.

The g.f. for the k-th column, (with leading zeros, and k>=5) is G(k,x)= x^ceiling(k/5)*P(k,x)/product(1-fallfac(p,5)*x,p=5..k), with fallfac(n,m) := A008279(n,m) (falling factorials), and P(k,x) := sum(A090222(k,m)*x^m,m=0..kmax(k)), k>=5, with kmax(k) := floor(4*(k-5)/5)= A090223(k-5). For the recurrence of the G(k,x) see A090222.

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 5 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, 5)^n, p=5..k), with fallfac(p, 5) := A008279(p, 5)=product(p+1-q, q=1..5); 5<= k <= 5*n, n>=1, else 0. From eq.(19) with r=5 of the Blasiak et al. reference.

[1]; [120,600,600,200,25,1]; [14400,504000,2664000,4608000,3501000,1350360,284800,33800,2225,75,1]; ...

Cf. A090217, A090209 (row sums), A090218 (alternating row sums).

Adjacent sequences: A090213 A090214 A090215 this_sequence A090217 A090218 A090219

Sequence in context: A067915 A115619 A100145 this_sequence A113546 A114887 A069085

nonn,easy,tabf

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