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%I #13 Apr 18 2013 00:30:29
%S 1,1,2,5,9,28,48,145,250,831,1404,4664,7875,26748,44960,154265,258777,
%T 896644,1501060,5239975,8758640,30760060,51350784,181258264,302271736,
%U 1071490551,1785262500,6351444132,10574365725,37738804488,62788919872
%N G.f. satisfies: a(2*n) equals coefficient of x^n in A(x)^(n+1) and a(2*n+1) equals coefficient of x^(n+1) in A(x)^(n+1), for n>=0, with a(0)=1.
%F a(2*n) = (n+1)*A094601(n) for n>=0.
%F Sum_{n>=1} a(2*n-1)*x^n/n = log(G(x)), where G(x) is the g.f. of A094601.
%F G.f. A(x) satisfies: A(x) = A(x*G(x)), where G(x) is the g.f. of A094601.
%F G.f.: A(x) = G(x^2) + x*G'(x^2)/G(x^2) + x^2*G'(x^2) where G(x) = (1/x)*Series_Reversion(x/A(x)) is the g.f. of A094601.
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 28*x^5 + 48*x^6 + 145*x^7 +...
%e Terms are formed from main and adjacent diagonals in the table of successive self-convolutions of this sequence:
%e [(1),(1), 2, 5, 9, 28, 48, 145, 250, 831, 1404, 4664,...];
%e [1,(2),(5), 14, 32, 94, 213, 588, 1343, 3726,...];
%e [1, 3,(9),(28), 75, 225, 590, 1656, 4287, 11780,...];
%e [1, 4, 14,(48),(145), 456, 1318, 3864, 10824, 30684,...];
%e [1, 5, 20, 75,(250),(831), 2590, 7980, 23755, 70155,...];
%e [1, 6, 27, 110, 399,(1404),(4664), 15102, 47355, 145880,...];
%e [1, 7, 35, 154, 602, 2240,(7875),(26748), 87892, 282093,...];
%e [1, 8, 44, 208, 870, 3416, 12648,(44960),(154265), 514920,...];
%e [1, 9, 54, 273, 1215, 5022, 19512, 72423,(258777),(896644),...]; ...
%e from which A094601 may be formed from the main diagonal:
%e [1/1, 2/2, 9/3, 48/4, 250/5, 1404/6, 7875/7, 44960/8, 258777/9,...].
%e Let G(x) be the g.f. of A094601:
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 50*x^4 + 234*x^5 + 1125*x^6 + 5620*x^7 +...
%e then the logarithm begins:
%e log(G(x)) = x + 5*x^2/2 + 28*x^3/3 + 145*x^4/4 + 831*x^5/5 + 4664*x^6/6 +...
%e and is formed from the odd-indexed terms of this sequence.
%o (PARI) {a(n)=local(A=1+x+x*O(x^n),G);for(i=1,ceil(log(n+1)/log(2)), G=serreverse(x/A)/x; A=subst(G+x*G',x,x^2)+x*subst(G',x,x^2)/subst(G,x,x^2) +x*O(x^n));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A094557, A094601.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 13 2004
%E Entry revised by _Paul D. Hanna_, Apr 17 2013
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