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A187010 G.f. satisfies: A(x) = Sum_{n>=0} x^n*g_n(x)^n where g_n(x) = A(x) - x^n*g_n(x)^n for n>=0. 0

%I

%S 1,1,1,4,8,20,65,189,545,1672,5234,16483,52508,168948,547031,1782892,

%T 5851234,19308826,64012154,213130527,712361672,2389177656,8038552120,

%U 27125159211,91774118484,311265968741,1058099992873,3604394906225

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n*g_n(x)^n where g_n(x) = A(x) - x^n*g_n(x)^n for n>=0.

%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 8*x^4 + 20*x^5 + 65*x^6 +...

%e where A(x) = 1 + x*g_1(x) + x^2*g_2(x)^2 + x^3*g_3(x)^3 +...

%e such that g_n(x) = A(x) - x^n*g_n(x)^n for n>=0.

%e The coefficients in the functions g_n(x) for n=0..8 begin:

%e n=0: [0,1,1,4,8,20,65,189,545,1672,5234,16483,52508,168948,...];

%e n=1: [1,0,1,3,5,15,50,139,406,1266,3968,12515,39993,128955,...];

%e n=2: [1,1,0,2,7,16,47,143,415,1264,4005,12713,40484,130346,...];

%e n=3: [1,1,1,3,5,14,49,150,443,1350,4208,13270,42368,136668,...];

%e n=4: [1,1,1,4,7,16,55,157,450,1412,4436,13987,44817,144544,...];

%e n=5: [1,1,1,4,8,19,60,174,495,1507,4728,14878,47318,152338,...];

%e n=6: [1,1,1,4,8,20,64,183,524,1598,4976,15637,49703,159564,...];

%e n=7: [1,1,1,4,8,20,65,188,538,1644,5129,16098,51178,164384,...];

%e n=8: [1,1,1,4,8,20,65,189,544,1664,5198,16339,51954,166940,...]; ...

%e The coefficients in g_n(x)^n for n=0..8 begin:

%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=1: [1, 0, 1, 3, 5, 15, 50, 139, 406, 1266, ...];

%e n=2: [1, 2, 1, 4, 18, 46, 130, 408, 1229, 3770, ...];

%e n=3: [1, 3, 6, 16, 39, 102, 322, 1026, 3213, 10140, ...];

%e n=4: [1, 4, 10, 32, 95, 260, 798, 2496, 7691, 24404, ...];

%e n=5: [1, 5, 15, 50, 165, 506, 1605, 5190, 16610, 53505, ...];

%e n=6: [1, 6, 21, 74, 258, 846, 2805, 9384, 31068, 102916, ...];

%e n=7: [1, 7, 28, 105, 385, 1330, 4564, 15723, 53606, 182000,...];

%e n=8: [1, 8, 36, 144, 554, 2008, 7128, 25208, 88171, 306144,...]; ...

%e where the antidiagonal sums of the above table equals this sequence.

%e Note how g_n(x) satisfies: A(x) = g_n(x) + x^n*g_n(x)^n for n>=0.

%o (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)}

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 19 2011

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Last modified January 26 19:45 EST 2021. Contains 340442 sequences. (Running on oeis4.)