%I A122888
%S A122888 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,
%T A122888 162,84,32,8,1,1,5,20,70,220,630,1656,4014,8994,18654,35832,63750,
%U A122888 105024,160120,225696,293685,352074,387820,391232,359992,300664,226580
%N A122888 Triangle, read by rows, where 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.
%C A122888 T(n, k) is the number of strings of length k-1 on the alphabet {1, 2, 
               ..., n} such that between every two occurrences of a letter i there 
               is an occurrence of a letter strictly larger than i. For example, 
               for n = 3, k = 4 we have the strings 121, 131, 232 and the six permutations 
               of 123. - Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 
               06 2008
%H A122888 Art of Problem Solving forum, <a href="http://www.artofproblemsolving.com/
               Forum/viewtopic.php?t=203662">Strings on [n] with certain restrictions</
               a>.
%F A122888 T(n,k) = [x^k] F_n(x) where F_{n+1}(x) = F_n(x+x^2) for n>=0, with F_1(x) 
               = x+x^2 and F_0(x)=x.
%e A122888 Triangle begins:
%e A122888 1;
%e A122888 1, 1;
%e A122888 1, 2, 2, 1;
%e A122888 1, 3, 6, 9, 10, 8, 4, 1;
%e A122888 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
%e A122888 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
%e A122888 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
%e A122888 1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
%e A122888 1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...;
%e A122888 1, 9, 72, 540, 3864, 26628, 177744, 1153740, 7303164, 45179508,...;
%e A122888 1, 10, 90, 765, 6240, 49260, 378312, 2836548, 20817588,...; ...
%e A122888 Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros,
%e A122888 yields the g.f. of row k in the triangle A122890:
%e A122888 1;
%e A122888 0, 1;
%e A122888 0, 0, 2;
%e A122888 0, 0, 1, 5;
%e A122888 0, 0, 0, 10, 14;
%e A122888 0, 0, 0, 8, 70, 42;
%e A122888 0, 0, 0, 4, 160, 424, 132;
%e A122888 0, 0, 0, 1, 250, 1978, 2382, 429;
%e A122888 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430; ...
%e A122888 in which the main diagonal is the Catalan numbers
%e A122888 and the row sums form the factorials.
%o A122888 (PARI) {T(n,k)=local(F=x+x^2, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, 
               x, G)); return(polcoeff(G, k, x)))}
%Y A122888 Cf. A007018 (row sums), diagonals: A112317, A112319, A122887; A092123 
               (largest term in row); A122889 (antidiagonal sums); A122890 (related 
               triangle).
%Y A122888 Sequence in context: A055870 A088459 A007799 this_sequence A092113 A045995 
               A157654
%Y A122888 Adjacent sequences: A122885 A122886 A122887 this_sequence A122889 A122890 
               A122891
%K A122888 nonn,tabf
%O A122888 0,5
%A A122888 Paul D. Hanna (pauldhanna(AT)juno.com), Sep 18 2006