OFFSET
0,4
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 8*x^4 + 20*x^5 + 65*x^6 +...
where A(x) = 1 + x*g_1(x) + x^2*g_2(x)^2 + x^3*g_3(x)^3 +...
such that g_n(x) = A(x) - x^n*g_n(x)^n for n>=0.
The coefficients in the functions g_n(x) for n=0..8 begin:
n=0: [0,1,1,4,8,20,65,189,545,1672,5234,16483,52508,168948,...];
n=1: [1,0,1,3,5,15,50,139,406,1266,3968,12515,39993,128955,...];
n=2: [1,1,0,2,7,16,47,143,415,1264,4005,12713,40484,130346,...];
n=3: [1,1,1,3,5,14,49,150,443,1350,4208,13270,42368,136668,...];
n=4: [1,1,1,4,7,16,55,157,450,1412,4436,13987,44817,144544,...];
n=5: [1,1,1,4,8,19,60,174,495,1507,4728,14878,47318,152338,...];
n=6: [1,1,1,4,8,20,64,183,524,1598,4976,15637,49703,159564,...];
n=7: [1,1,1,4,8,20,65,188,538,1644,5129,16098,51178,164384,...];
n=8: [1,1,1,4,8,20,65,189,544,1664,5198,16339,51954,166940,...]; ...
The coefficients in g_n(x)^n for n=0..8 begin:
n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 0, 1, 3, 5, 15, 50, 139, 406, 1266, ...];
n=2: [1, 2, 1, 4, 18, 46, 130, 408, 1229, 3770, ...];
n=3: [1, 3, 6, 16, 39, 102, 322, 1026, 3213, 10140, ...];
n=4: [1, 4, 10, 32, 95, 260, 798, 2496, 7691, 24404, ...];
n=5: [1, 5, 15, 50, 165, 506, 1605, 5190, 16610, 53505, ...];
n=6: [1, 6, 21, 74, 258, 846, 2805, 9384, 31068, 102916, ...];
n=7: [1, 7, 28, 105, 385, 1330, 4564, 15723, 53606, 182000,...];
n=8: [1, 8, 36, 144, 554, 2008, 7128, 25208, 88171, 306144,...]; ...
where the antidiagonal sums of the above table equals this sequence.
Note how g_n(x) satisfies: A(x) = g_n(x) + x^n*g_n(x)^n for n>=0.
PROG
(PARI) {a(n)=local(G=1, g=vector(n+1, k, 1+x+x*O(x^n))); for(i=1, n, G=1+sum(m=1, n, x^m*g[m+1]^m); g=vector(n+1, k, G-x^(k-1)*g[k]^(k-1)+x*O(x^n)); ); polcoeff(G, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 19 2011
STATUS
approved