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%I #11 May 07 2016 07:59:53
%S 1,1,2,6,20,72,274,1084,4413,18365,77773,334065,1451929,6373379,
%T 28214851,125826137,564736106,2549004457,11563003842,52688331678,
%U 241049183598,1106818411361,5098961600041,23561122837102,109171473224810,507136264223327,2361332083301582,11018704225690520,51520180227572593,241344269535483201,1132543412040153639,5323310040423904420
%N G.f. A(x) satisfies: A(x)^3 = x * A( A(x)^2 + A(x)^3 ).
%H Paul D. Hanna, <a href="/A272485/b272485.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies: A( x^3 / A(x^2 + x^3) ) = x.
%F a(n) ~ c * d^n / n^(3/2), where d = 4.93477240448054430929819... and c = 0.06099488383521219235235... . - _Vaclav Kotesovec_, May 07 2016
%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 72*x^6 + 274*x^7 + 1084*x^8 + 4413*x^9 + 18365*x^10 + 77773*x^11 + 334065*x^12 +...
%e where A(x)^3 = x * A( A(x)^2 + A(x)^3 ).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 208*x^7 + 808*x^8 + 3244*x^9 + 13354*x^10 + 56060*x^11 + 239080*x^12 + 1032908*x^13 + 4511152*x^14 + 19884256*x^15 +...
%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 438*x^8 + 1742*x^9 + 7116*x^10 + 29685*x^11 + 125937*x^12 + 541704*x^13 + 2357028*x^14 + 10355984*x^15 +...
%e A(x^2 + x^3) = x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 6*x^7 + 12*x^8 + 26*x^9 + 56*x^10 + 124*x^11 + 278*x^12 + 632*x^13 + 1454*x^14 + 3378*x^15 + 7918*x^16 + 18694*x^17 + 44427*x^18 + 106175*x^19 + 255031*x^20 + 615320*x^21 + 1490588*x^22 +...
%e Let B(x) be the series reversion of g.f. A(x), then
%e B(x) = x - x^2 - x^4 - 2*x^6 - 2*x^7 - 7*x^8 - 11*x^9 - 28*x^10 - 57*x^11 - 134*x^12 - 300*x^13 - 700*x^14 - 1627*x^15 - 3833*x^16 - 9075*x^17 - 21628*x^18 - 51818*x^19 - 124727*x^20 - 301523*x^21 - 731662*x^22 - 1781632*x^23 +...
%e where B(x) = x^3 / A(x^2 + x^3).
%o (PARI) /* From definition A( A(x)^2 + A(x)^3 ) = A(x)^3/x */
%o {a(n) = my(A=[1, 1,0]); for(i=1, n, A = concat(A, 0); G=x*Ser(A); A[#A-1] = Vec(x*subst(G, x, G^2 + G^3)/G^3)[#A-1] ); A[n]}
%o for(n=1, 40, print1(a(n), ", "))
%o (PARI) /* Using Series Reversion of x^3/A(x^2+x^3) */
%o {a(n) = my(A=x); for(i=1, n, A = serreverse( x^3/subst(A, x, x^2 + x^3 +x^2*O(x^n))) ); polcoeff(A, n)}
%o for(n=1, 40, print1(a(n), ", "))
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 06 2016
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