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%I #14 Aug 28 2017 07:42:08
%S 1,1,2,6,16,48,152,501,1690,5822,20388,72360,259688,940792,3435904,
%T 12636554,46760376,173971252,650380288,2441905192,9203979808,
%U 34813551616,132101846848,502732914346,1918353118348,7338208929260,28134551443480,108094972590872,416122805092224,1604832481200352,6199797669769760,23989294121910790,92962226232374892,360749306397285812
%N G.f. satisfies: A(x - A(x) + A(x)^2) = -x^4.
%C At what positions n is a(n) odd?
%C Compare g.f. to: C(x - C(x) + C(x)^2) = 0, trivial when C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A291189/b291189.txt">Table of n, a(n) for n = 1..512</a>
%F G.f. A(x) satisfies: x - A(x) + A(x)^2 = Ai(-x^4) where Ai( A(x) ) = x.
%F a(n) ~ c * d^n / n^(3/2), where d = 4.05999022767846206402248334679744980701174... and c = 0.14415462031792796731396571657... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 48*x^6 + 152*x^7 + 501*x^8 + 1690*x^9 + 5822*x^10 + 20388*x^11 + 72360*x^12 + 259688*x^13 + 940792*x^14 + 3435904*x^15 + 12636554*x^16 + 46760376*x^17 + 173971252*x^18 + 650380288*x^19 + 2441905192*x^20 + 9203979808*x^21 +...
%e where A(x - A(x) + A(x)^2) = -x^4.
%e RELATED SERIES.
%e Define Ai(x) such that Ai(A(x)) = x, where Ai(x) begins:
%e Ai(x) = x - x^2 - x^4 + 4*x^5 - 6*x^6 + 8*x^7 - 30*x^8 + 92*x^9 - 190*x^10 + 428*x^11 - 1276*x^12 + 3524*x^13 - 8572*x^14 + 22120*x^15 - 62215*x^16 + 169464*x^17 - 444860*x^18 + 1202364*x^19 - 3340582*x^20 + 9167812*x^21 - 24936852*x^22 + 68746520*x^23 - 191319986*x^24 + 530404940*x^25 +...
%e then x - A(x) + A(x)^2 = Ai(-x^4),
%e and Ai(x) - Ai( -Ai(x)^4 ) = x - x^2.
%o (PARI) {a(n) = my(A=x,V=[1, 1, 2,6]); for(i=1,n, V=concat(V,0); A=x*Ser(V); V[#V]=Vec(subst(A,x,x - A + A^2))[#V-3]);V[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A291190.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Aug 20 2017
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