%I A128520
%S A128520 1,1,1,2,1,4,2,3,1,6,8,8,7,3,5,1,8,18,21,28,17,23,14,11,5,8,1,10,32,50,
%T A128520 73,73,77,79,58,57,44,39,22,18,8,13,1,12,50,103,166,221,238,289,256,269,
%U A128520 226,231,176,166,113,119,74,62,36,29,13,21,1,14,72,188,347,545,680,861
%N A128520 q-inverse of x-x^2-x^3. Coefficients of a solution to the functional 
               equation x = f(x) - f(x) f(q x) - f(x) f(q x) f(q^2 x).
%C A128520 The n-th row has binomial(n-1,2)+1 entries for n>=1. Each 1 starts a 
               new row.
%C A128520 The last entry in the n-th row is the Fibonacci number F(n) = A000045(n). 
               The second to last entry is F(n-1). The third from the end is the 
               Lucas number L(n-1) = A000032(n-1).
%C A128520 The first entry in the n-th row is 1: A000012. The second entry is 2(n-2) 
               = A005843(n-2). The third entry is 2(n-3)^2 = A001105(n-3).
%C A128520 The n-th row sum is A001002(n-1).
%F A128520 Let a(n,i) be the i-th entry in the n-th row, for n >= 1 and 1 <= i <= 
               binomial(n-1,2)+1. Let f_n(q) = Sum_{i=1..binomial(n-1,2)+1} a(n,
               i) q^(n+i-2) and f(x) = Sum_{n>=1} f_n(q) x^n. Then f satisfies x 
               = f(x) - f(x) f(q x) - f(x) f(q x) f(q^2 x).
%F A128520 The functions f_n satisfy the recurrence f_1 = 1, f_n = Sum_{i=1..n-1}(q^i 
               f_i f_{n-i}) + Sum_{i=1..n-2}( f_i Sum_{j=1..n-i-1}(q^(2i+j) f_j 
               f_{n-i-j})).
%F A128520 Equivalently, let g_n(q) = f_n(q)/q^(n-1) = Sum_{i=0..binomial(n-1,2)} 
               a(n,i) q^(i-2) and g(x) = q f(x/q) = Sum_{n>=1} g_n(q) x^n. Then 
               g satisfies q^2 x = q^2 g(x) - q g(x) g(q x) - g(x) g(q x) g(q^2 
               x). The functions g_n satisfy g_1 = 1, g_n = Sum_{i=1..n-1} (q^(i-1) 
               g_i g_{n-i}) + Sum_{i=1..n-2} Sum_{j=1..n-i-1} (q^(2i+j-2) g_i g_j 
               g_{n-i-j}).
%e A128520 1; 1; 1, 2; 1, 4, 2, 3; 1, 6, 8, 8, 7, 3, 5; 1, 8, 18, 21, 28, 17, 23, 
               14, 11, 5, 8; ...
%e A128520 a(4,1)=1, a(4,2)=4, a(4,3)=2, a(4,4)=3. f_4(q) = q^3 + 4 q^4 + 2 q^5 
               + 3 q^6. g_4(q) = 1 + 4 q + 2 q^2 + 3 q^3.
%p A128520 a:= proc(n)local i, j; option remember; if n=1 then 1 else add(q^i*a(i)*a(n-i), 
               i=1..n-1) + add(a(i) *add(q^(2*n-2*i-j)*a(j)*a(n-i-j), j=1..n-i-1), 
               i=1..n-2); fi; expand(%); sort(%); end;
%t A128520 g[1]=1; g[n_]:=g[n]=Expand[Sum[q^(i-1)g[i]g[n-i],{i,1,n-1}]+Sum[q^(2i+j-2)g[i]g[j]g[n-i-j],
               {i,1,n-2},{j,1,n-i-1}]]; a[n_,i_]:=Coefficient[g[n],q,i-1]
%Y A128520 Cf. A001002, A000045, A000032, A005843, A000012, A001105.
%Y A128520 Sequence in context: A078458 A033317 A007733 this_sequence A123755 A118291 
               A118290
%Y A128520 Adjacent sequences: A128517 A128518 A128519 this_sequence A128521 A128522 
               A128523
%K A128520 nonn,tabf
%O A128520 1,4
%A A128520 Brian Drake (bdrake(AT)brandeis.edu), Mar 06 2007
%E A128520 Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Mar 09 2007