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A375438 Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.

%I A375438 #9 Aug 22 2024 04:21:48
%S A375438 1,1,1,1,2,4,6,12,24,46,93,191,393,819,1724,3648,7772,16654,35850,
%T A375438 77520,168289,366629,801328,1756620,3861157,8508247,18791480,41591566,
%U A375438 92237598,204931918,456096480,1016720162,2269865456,5074732028,11360680664,25464831258,57146836290
%N A375438 Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.
%H A375438 Paul D. Hanna, <a href="/A375438/b375438.txt">Table of n, a(n) for n = 1..1000</a>
%F A375438 a(n) ~ c * d^n / n^(3/2), where d = 2.3394011548205191342723840863090815005163727107... and c = 0.278795592719328257510209001410993009380027455... - _Vaclav Kotesovec_, Aug 22 2024
%e A375438 G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 24*x^9 + 46*x^10 + 93*x^11 + 191*x^12 + 393*x^13 + 819*x^14 + 1724*x^15 + ...
%e A375438 where A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.
%e A375438 RELATED SERIES.
%e A375438 A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 36*x^8 + 70*x^9 + 138*x^10 + 279*x^11 + 571*x^12 + 1179*x^13 + 2457*x^14 + 5168*x^15 + ...
%e A375438 Let B(x) be the series reversion, B(A(x)) = x, then B(x) begins
%e A375438 B(x) = x - x^2 + x^3 - x^4 + 3*x^6 - 8*x^7 + 9*x^8 + 18*x^9 - 134*x^10 + 442*x^11 - 997*x^12 + 1428*x^13 - 10*x^14 - 7640*x^15 + ...
%e A375438 SPECIFIC VALUES.
%e A375438 A(2/5) = 0.741461459188681119672668058998130332678610537393868...
%e A375438 A(1/3) = 0.515838591521774544528452689654484632143493145820237...
%e A375438 A(1/4) = 0.335691575266570204286454430830296229544471248787335...
%e A375438 A(1/5) = 0.250623759777806277498640241328479184446996870792060...
%e A375438 A(1/6) = 0.200220725056085320106333620370114891484630832364434...
%o A375438 (PARI) {a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);
%o A375438 A[#A] = polcoeff( x + x^2 + ( Ax^3 + 2*subst(Ax,x,x^3))/3 - Ax,#A-1) );A[n+1]}
%o A375438 for(n=1,40,print1(a(n),", "))
%Y A375438 Cf. A375439, A357538, A000625.
%K A375438 nonn
%O A375438 1,5
%A A375438 _Paul D. Hanna_, Aug 21 2024