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