1,2

The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.

The g.f. for the k-th column, (with leading zeros and k>=3) is G(k,x)= x^ceiling(k/3)*P(k,x)/product(1-fallfac(p,3)*x,p=3..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A089517(k,m)*x^m,m=0..kmax(k)), k>=3, with kmax(k) := A004523(k-3)= floor(2*(k-3)/3)= [0,0,1,2,2,3,4,4,5,...]. For the recurrence of the G(k,x) see A089517. Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003

For the computation of the k-th column sequence see A090219.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers

W. Lang, First 6 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, 3)^n, p=3..k), with fallfac(p, 3) := A008279(p, 3)=p*(p-1)*(p-2); 3<= k <= 3*n, n>=1, else 0. From eq.(19) with r=3 of the Blasiak et al. reference.

1; 6,18,9,1; 36,540,1242,882,243,27,1; ...

Row sums give A069223. Cf. A078739.

The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.

A089504, A069223 (row sums), A090212 (alternating row sums).

Sequence in context: A077022 A074923 A093061 this_sequence A129870 A091014 A097370

Adjacent sequences: A078738 A078739 A078740 this_sequence A078742 A078743 A078744

nonn,tabf,easy

N. J. A. Sloane (njas(AT)research.att.com), Dec 21 2002