%I A122890
%S A122890 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,
%T A122890 0,1,250,1978,2382,429,0,0,0,0,302,6276,19508,12804,1430,0,0,0,0,298,
%U A122890 15674,106492,168608,66946,4862,0,0,0,0,244,33148,451948,1445208
%N A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields 
               the g.f. of column n in the triangle A122888, for n>=1.
%C A122890 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.
%F A122890 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 11 2009: 
               (Start)
%F A122890 G.f. of row n = (1-x)^n*[g.f. of column n of A122888] where
%F A122890 the g.f. of row n of A122888 is the n-th iteration of x+x^2.
%F A122890 ...
%F A122890 Row-reversal forms triangle A158830 where
%F A122890 g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825], and
%F A122890 the g.f. of row n of array A158825 is the n-th iteration of x*C(x)
%F A122890 and C(x) is the g.f. of the Catalan sequence A000108.
%F A122890 (End)
%e A122890 Triangle begins:
%e A122890 .1;
%e A122890 .0,1;
%e A122890 .0,0,2;
%e A122890 .0,0,1,5;
%e A122890 .0,0,0,10,14;
%e A122890 .0,0,0,8,70,42;
%e A122890 .0,0,0,4,160,424,132;
%e A122890 .0,0,0,1,250,1978,2382,429;
%e A122890 .0,0,0,0,302,6276,19508,12804,1430;
%e A122890 .0,0,0,0,298,15674,106492,168608,66946,4862;
%e A122890 .0,0,0,0,244,33148,451948,1445208,1337684,343772,16796;
%e A122890 .0,0,0,0,162,61806,1614906,9459090,16974314,10003422,1744314,58786;
%e A122890 .0,0,0,0,84,103932,5090124,51436848,161380816,180308420,71692452,8780912,
               208012;
%e A122890 Table A122888 starts:
%e A122890 .1;
%e A122890 .1, 1;
%e A122890 .1, 2, 2, 1;
%e A122890 .1, 3, 6, 9, 10, 8, 4, 1;
%e A122890 .1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
%e A122890 .1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
%e A122890 .1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
%e A122890 where row n gives the g.f. of the n-th self-composition of (x+x^2).
%e A122890 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 11 2009: 
               (Start)
%e A122890 ROW-REVERSAL yields triangle A158830:
%e A122890 .1;
%e A122890 .1,0;
%e A122890 .2,0,0;
%e A122890 .5,1,0,0;
%e A122890 .14,10,0,0,0;
%e A122890 .42,70,8,0,0,0;
%e A122890 .132,424,160,4,0,0,0;
%e A122890 .429,2382,1978,250,1,0,0,0; ...
%e A122890 where
%e A122890 g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825];
%e A122890 g.f. of row n of A158825 = n-th iteration of x*Catalan(x).
%e A122890 RELATED ARRAY A158825 begins:
%e A122890 .1,1,2,5,14,42,132,429,1430,4862,16796,58786,...;
%e A122890 .1,2,6,21,80,322,1348,5814,25674,115566,528528,...;
%e A122890 .1,3,12,54,260,1310,6824,36478,199094,1105478,...;
%e A122890 .1,4,20,110,640,3870,24084,153306,993978,...;
%e A122890 .1,5,30,195,1330,9380,67844,500619,3755156,...;
%e A122890 .1,6,42,315,2464,19852,163576,1372196,11682348,...;
%e A122890 .1,7,56,476,4200,38052,351792,3305484,31478628,...;
%e A122890 .1,8,72,684,6720,67620,693048,7209036,75915708,...; ...
%e A122890 which consists of successive iterations of x*Catalan(x).
%e A122890 (End)
%Y A122890 Cf. A122888; A122891 (column sums); diagonals: A122892, A000108.
%Y A122890 Cf. related tables: A158830, A158825. [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Apr 11 2009]
%Y A122890 Sequence in context: A137585 A072458 A067310 this_sequence A138497 A113129 
               A127826
%Y A122890 Adjacent sequences: A122887 A122888 A122889 this_sequence A122891 A122892 
               A122893
%K A122890 nonn,tabl
%O A122890 0,6
%A A122890 Paul D. Hanna (pauldhanna(AT)juno.com), Sep 18 2006