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%I #19 Jan 09 2021 02:11:47
%S 1,1,3,16,121,1166,13508,181616,2769085,47109136,883639920,
%T 18107148225,402374973506,9637795235498,247545497301618,
%U 6787859682565805,197933312666622029,6116627434424525916,199695744998600583128,6868763166262716491823,248281343442023732418598
%N G.f. satisfies: A(x) = 1 + x*A(x) * A(x*A(x)) + x^2*A(x)^2 * A'(x*A(x)).
%F a(n) = A113882(n+1) / (n+1) for n>=0.
%F G.f. A(x) satisfies:
%F (1) A(x) = (1/x)*Series_Reversion(x/(1 + x*A(x) + x^2*A'(x))).
%F (2) A(x) = (1/x)*Series_Reversion(x/G(x)) = G(x*A(x)) where G(x) = 1 + x*A(x) + x^2*A'(x) = A(x/G(x)) is the g.f. of A113882 (number of well-nested drawings of a rooted tree).
%e G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 121*x^4 + 1166*x^5 + 13508*x^6 + ...
%e Related expansions:
%e A(x*A(x)) = 1 + x + 4*x^2 + 25*x^3 + 206*x^4 + 2077*x^5 + 24603*x^6 + ...
%e A'(x*A(x)) = 1 + 6*x + 54*x^2 + 598*x^3 + 7714*x^4 + 112726*x^5 + ...
%e A(x)*A(A(x)) = 1 + 2*x + 8*x^2 + 48*x^3 + 380*x^4 + 3709*x^5 + 42856*x^6 + ...
%e A(x)^2*A'(A(x)) = 1 + 8*x + 73*x^2 + 786*x^3 + 9799*x^4 + 138760*x^5 + ...
%e where A(x) = 1 + x*A(x)*A(x*A(x)) + x^2*A(x)^2*A'(x*A(x)).
%e Let G(x) = 1 + x*A(x) + x^2*A'(x) be the g.f. of A113882;
%e the table of coefficients in G(x)^n begins:
%e G(x)^1: [(1), 1, 2, 9, 64, 605, 6996, 94556, ...];
%e G(x)^2: [1, (2), 5, 22, 150, 1374, 15539, 206676, ...];
%e G(x)^3: [1, 3, (9), 40, 264, 2346, 25937, 339294, ...];
%e G(x)^4: [1, 4, 14, (64), 413, 3568, 38558, 495848, ...];
%e G(x)^5: [1, 5, 20, 95, (605), 5096, 53840, 680365, ...];
%e G(x)^6: [1, 6, 27, 134, 849, (6996), 72302, 897558, ...];
%e G(x)^7: [1, 7, 35, 182, 1155, 9345, (94556), 1152936, ...]; ...
%e where the terms in parentheses form the initial terms of this sequence:
%e [(1)/1, (2)/2, (9)/3, (64)/4, (605)/5, (6996)/6, (94556)/7, ...].
%e Further, the logarithm of the g.f. A(x) may be formed from a diagonal in the above table:
%e log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 413*x^4/4 + 5096*x^5/5 + 72302*x^6/6 + 1152936*x^7/7 + ...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*subst(A,x,x*A+x*O(x^n))+x^2*A^2*subst(A',x,x*A+x*O(x^n)));polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) {a(n)=local(G=1+x);for(k=0,n,G=1+x*deriv(serreverse(x/(G+x^2*O(x^n))))); polcoeff(serreverse(x/G)/x,n)}
%Y Cf. A113882.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 08 2007
%E More terms and name changed by _Paul D. Hanna_, Apr 28 2012
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