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%I #27 Mar 19 2016 10:32:11
%S 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 162,84,32,8,1,1,5,20,70,220,630,1656,4014,8994,18654,35832,63750,
%U 105024,160120,225696,293685,352074,387820,391232,359992,300664,226580
%N Triangle, read by rows, where row n lists the coefficients of x^k, k=1..2^n, in the n-th iteration of (x + x^2) for n>=0.
%C 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 B. Lewis_, May 06 2008
%H Paul D. Hanna, <a href="/A122888/b122888.txt">Table of n, a(n) for n = 0..2046</a> (rows 0..10)
%H Art of Problem Solving forum, <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=203662">Strings on [n] with certain restrictions</a>.
%F T(n,k) = [x^k] F_n(x) where F_{n+1}(x) = F_n(x+x^2) for n>=1, with F_0(x)=x.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 2, 1;
%e 1, 3, 6, 9, 10, 8, 4, 1;
%e 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
%e 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
%e 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
%e 1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
%e 1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...;
%e 1, 9, 72, 540, 3864, 26628, 177744, 1153740, 7303164, 45179508,...;
%e 1, 10, 90, 765, 6240, 49260, 378312, 2836548, 20817588,...; ...
%e Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros,
%e yields the g.f. of row k in the triangle A122890:
%e 1;
%e 0, 1;
%e 0, 0, 2;
%e 0, 0, 1, 5;
%e 0, 0, 0, 10, 14;
%e 0, 0, 0, 8, 70, 42;
%e 0, 0, 0, 4, 160, 424, 132;
%e 0, 0, 0, 1, 250, 1978, 2382, 429;
%e 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430; ...
%e in which the main diagonal is the Catalan numbers
%e and the row sums form the factorials.
%p b:= proc(n) option remember; `if`(n=0, x,
%p expand((x-> x+x^2)(b(n-1))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p seq(T(n), n=0..5); # _Alois P. Heinz_, Mar 14 2016
%t f[0][x_] = x; f[n_][x_] := f[n][x] = f[n-1][x+x^2]; row[n_] := CoefficientList[f[n][x], x] // Rest; Table[row[n], {n, 0, 5} ] // Flatten (* _Jean-François Alcover_, Sep 10 2012 *)
%o (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)))}
%o for(n=0, 6, for(k=1, 2^n, print1(T(n, k), ", ")); print(""))
%o (Maxima) T(m,n):=if m=0 and n=1 then 1 else if m=0 and n>1 then 0 else if m=1 then binomial(1,n-1) else sum(binomial(i,n-i)*T(m-1,i),i,1,n); [_Vladimir Kruchinin_, May 19 2012]
%Y Cf. A007018 (row sums), diagonals: A112317, A112319, A122887; A092123 (largest term in row); A122889 (antidiagonal sums); A122890 (related triangle).
%K nonn,tabf
%O 0,5
%A _Paul D. Hanna_, Sep 18 2006
%E Name changed slightly by _Paul D. Hanna_, Apr 29 2013
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