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.

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 11 2009: (Start)

G.f. of row n = (1-x)^n*[g.f. of column n of A122888] where

the g.f. of row n of A122888 is the n-th iteration of x+x^2.

...

Row-reversal forms triangle A158830 where

g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825], and

the g.f. of row n of array A158825 is the n-th iteration of x*C(x)

and C(x) is the g.f. of the Catalan sequence A000108.

(End)

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).

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 11 2009: (Start)

ROW-REVERSAL yields triangle A158830:

.1;

.1,0;

.2,0,0;

.5,1,0,0;

.14,10,0,0,0;

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

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

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

where

g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825];

g.f. of row n of A158825 = n-th iteration of x*Catalan(x).

RELATED ARRAY A158825 begins:

.1,1,2,5,14,42,132,429,1430,4862,16796,58786,...;

.1,2,6,21,80,322,1348,5814,25674,115566,528528,...;

.1,3,12,54,260,1310,6824,36478,199094,1105478,...;

.1,4,20,110,640,3870,24084,153306,993978,...;

.1,5,30,195,1330,9380,67844,500619,3755156,...;

.1,6,42,315,2464,19852,163576,1372196,11682348,...;

.1,7,56,476,4200,38052,351792,3305484,31478628,...;

.1,8,72,684,6720,67620,693048,7209036,75915708,...; ...

which consists of successive iterations of x*Catalan(x).

(End)

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

Cf. related tables: A158830, A158825. [From Paul D. Hanna (pauldhanna(AT)juno.com), Apr 11 2009]

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