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%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
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