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A242795 a(n) = [x^n] ( 1 + x*A(x)^(n+1) )^(n+1) / (n+1) for n>=0, with a(0)=1.

%I A242795 #5 May 22 2014 22:33:28
%S A242795 1,1,4,35,466,8072,168330,4045603,109089538,3242538284,104946776716,
%T A242795 3665946814257,137291732981170,5483948111154008,232660368810666229,
%U A242795 10447887814670412307,495139557493903545618,24699919104193662112382,1293980412831022073519196
%N A242795 a(n) = [x^n] ( 1 + x*A(x)^(n+1) )^(n+1) / (n+1) for n>=0, with a(0)=1.
%C A242795  Compare to the g.f. G(x) = x + x*G(G(x)) of A030266 that satisfies:
%C A242795 A030266(n+1) = [x^n] ( 1 + G(x) )^(n+1) / (n+1) for n>=0.
%e A242795 G.f.: A(x) = 1 + x + 4*x^2 + 35*x^3 + 466*x^4 + 8072*x^5 + 168330*x^6 +...
%e A242795 Form a table of coefficients of x^k in (1 + x*A(x)^(n+1))^(n+1) like so:
%e A242795 n=0: [1,  1,   1,    4,    35,    466,    8072,   168330, ...];
%e A242795 n=1: [1,  2,   5,   22,   178,   2228,   37141,   755702, ...];
%e A242795 n=2: [1,  3,  12,   64,   516,   6126,   97725,  1929927, ...];
%e A242795 n=3: [1,  4,  22,  140,  1177,  13548,  206876,  3946612, ...];
%e A242795 n=4: [1,  5,  35,  260,  2330,  26626,  391830,  7202170, ...];
%e A242795 n=5: [1,  6,  51,  434,  4185,  48432,  694714, 12312642, ...];
%e A242795 n=6: [1,  7,  70,  672,  6993,  83174, 1178310, 20224653, ...];
%e A242795 n=7: [1,  8,  92,  984, 11046, 136392, 1932876, 32364824, ...];
%e A242795 n=8: [1,  9, 117, 1380, 16677, 215154, 3084024, 50833962, ...];
%e A242795 n=9: [1, 10, 145, 1870, 24260, 328252, 4801655, 78652350, ...]; ...
%e A242795 then this sequence is formed from the main diagonal:
%e A242795 [1/1, 2/2, 12/3, 140/4, 2330/5, 48432/6, 1178310/7, 32364824/8, ...].
%o A242795 (PARI) {a(n)=local(A=[1,1]);for(m=1,n,A=concat(A,0);A[m+1]=Vec((1+x*Ser(A)^(m+1))^(m+1))[m+1]/(m+1));A[n+1]}
%o A242795 for(n=0,25,print1(a(n),", "))
%Y A242795 Cf. A242794.
%K A242795 nonn
%O A242795 0,3
%A A242795 _Paul D. Hanna_, May 22 2014