A242794 - OEIS

%I #6 May 22 2014 22:32:42

%S 1,1,3,22,257,3986,75304,1653086,40979297,1126004203,33856704386,

%T 1103686134563,38734891315775,1455569736467094,58304721086789654,

%U 2480233978808257526,111686585878084164913,5308774844414927594856,265682854185812938555354,13966882165871163036529423

%N a(n) = [x^n] ( 1 + x*A(x)^n )^(n+1) / (n+1) for n>=0, with a(0)=1.

%C  Compare to the g.f. G(x) = x + x*G(G(x)) of A030266 that satisfies:

%C A030266(n+1) = [x^n] ( 1 + G(x) )^(n+1) / (n+1) for n>=0.

%e G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 257*x^4 + 3986*x^5 + 75304*x^6 +...

%e Form a table of coefficients of x^k in (1 + x*A(x)^n)^(n+1) like so:

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

%e n=1: [1, 2,   3,    8,    51,    564,    8539,   159226, ...];

%e n=2: [1, 3,   9,   34,   210,   2118,   30245,   544962, ...];

%e n=3: [1, 4,  18,   88,   575,   5472,   73242,  1263604, ...];

%e n=4: [1, 5,  30,  180,  1285,  12016,  151820,  2490390, ...];

%e n=5: [1, 6,  45,  320,  2520,  23916,  290162,  4518600, ...];

%e n=6: [1, 7,  63,  518,  4501,  44310,  527128,  7834548, ...];

%e n=7: [1, 8,  84,  784,  7490,  77504,  922096, 13224688, ...];

%e n=8: [1, 9, 108, 1128, 11790, 129168, 1561860, 21921156, ...]; ...

%e then this sequence is formed from the main diagonal:

%e [1/1, 2/2, 9/3, 88/4, 1285/5, 23916/6, 527128/7, 13224688/8, ...].

%o (PARI) {a(n)=local(A=[1,1]);for(m=1,n,A=concat(A,0);A[m+1]=Vec((1+x*Ser(A)^m)^(m+1))[m+1]/(m+1));A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A242795.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 22 2014