0,6

Main diagonal forms the Catalan numbers (A000108). Row sums gives the factorials. In table A122888, row n lists the coefficients of x^k, k=1..2^n, in the n-th self-composition of (x + x^2) for n>=0.

Triangle begins:

1;

0,1;

0,0,2;

0,0,1,5;

0,0,0,10,14;

0,0,0,8,70,42;

0,0,0,4,160,424,132;

0,0,0,1,250,1978,2382,429;

0,0,0,0,302,6276,19508,12804,1430;

0,0,0,0,298,15674,106492,168608,66946,4862;

0,0,0,0,244,33148,451948,1445208,1337684,343772,16796;

0,0,0,0,162,61806,1614906,9459090,16974314,10003422,1744314,58786;

0,0,0,0,84,103932,5090124,51436848,161380816,180308420,71692452,8780912,208012;

Table A122888 starts:

1;

1, 1;

1, 2, 2, 1;

1, 3, 6, 9, 10, 8, 4, 1;

1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;

1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;

1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;

where row n gives the g.f. of the n-th self-composition of (x+x^2).

Cf. A122888; A122891 (column sums); diagonals: A122892, A000108.

Sequence in context: A137585 A072458 A067310 this_sequence A138497 A113129 A127826

Adjacent sequences: A122887 A122888 A122889 this_sequence A122891 A122892 A122893

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 18 2006