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%I #14 Aug 28 2023 10:51:40
%S 1,1,3,13,66,364,2116,12768,79222,502297,3240120,21196593,140295584,
%T 937787728,6321624862,42926227470,293350136170,2015999854478,
%U 13923926272607,96598395025615,672852440805930,4703751150849738,32991286060134402,232091541493091566
%N G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^8.
%H Vaclav Kotesovec, <a href="/A228987/b228987.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. satisfies: A(x) = 1/A(-x*A(x)^5); note that the function G(x) = 1 + x*G(x)^3 (g.f. of A001764) also satisfies this condition: G(x) = 1/G(-x*G(x)^5).
%F a(n) ~ sqrt((3 - r*s^2)/(2*Pi*(3 + 28*r^2*s^5))) / (4*n^(3/2)*r^(n + 1/2)), where r = 0.1331154541373089587498695338172936885734070972340... and s = 1.408602671059676188189711196409966797670750551605... are real roots of the system of equations 1 + r*s^3 + r^3*s^8 = s, 3*r*s^2 + 8*r^3*s^7 = 1. - _Vaclav Kotesovec_, Nov 22 2017
%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(3*n-k+1,n-2*k)/(3*n-k+1). - _Seiichi Manyama_, Aug 28 2023
%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 364*x^5 + 2116*x^6 +...
%e Related expansions:
%e A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 312*x^4 + 1788*x^5 + 10686*x^6 +...
%e A(x)^8 = 1 + 8*x + 52*x^2 + 328*x^3 + 2082*x^4 + 13384*x^5 + 87124*x^6 +...
%e Given (1) A(x) = 1 + x*A(x)^3 + x^3*A(x)^8,
%e suppose (2) A(x) = 1/A(-x*A(x)^5),
%e then substituting x in (1) with -x*A(x)^5 yields:
%e 1/A(x) = 1 - x*A(x)^5/A(x)^3 - x^3*A(x)^15/A(x)^8,
%e which illustrates that (2) is consistent with (1).
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3+x^3*A^8 +x*O(x^n)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A216490, A112806.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 10 2013
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