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Expansion of g.f. A(x) satisfying A( x*(1 + 3*x)*A(x)^2 )^5 = A( x^2*(1 + 5*x)*A(x)^3 )^3.
3

%I #11 Mar 08 2024 16:58:58

%S 1,0,-15,80,-255,432,1020,-12510,71595,-354070,1570104,-5622420,

%T 13438045,-4035840,-135362385,608528868,-646486560,-5733423180,

%U 27001866935,58064077890,-1517240012304,12354566895940,-72313804557375,336406302349200,-1168330262135925,1853669275129548

%N Expansion of g.f. A(x) satisfying A( x*(1 + 3*x)*A(x)^2 )^5 = A( x^2*(1 + 5*x)*A(x)^3 )^3.

%H Paul D. Hanna, <a href="/A370533/b370533.txt">Table of n, a(n) for n = 1..501</a>

%e G.f.: A(X) = x - 15*x^3 + 80*x^4 - 255*x^5 + 432*x^6 + 1020*x^7 - 12510*x^8 + 71595*x^9 - 354070*x^10 + 1570104*x^11 - 5622420*x^12 + ...

%e where A( x*(1 + 3*x)*A(x)^2 )^5 = A( x^2*(1 + 5*x)*A(x)^3 )^3.

%e RELATED SERIES.

%e B(x) = A( x*(1 + 3*x)*A^2 )^(1/3) = A( x^2*(1 + 5*x)*A(x)^3 )^(1/5)

%e where B(x) is the g.f. of A370534, which begins

%e B(x) = x + x^2 - 11*x^3 + 45*x^4 - 135*x^5 + 297*x^6 - 198*x^7 - 1872*x^8 + 15705*x^9 - 103695*x^10 + 556749*x^11 - 2275131*x^12 + 6904116*x^13 + ...

%e B(x)^3 = A( x*(1 + 3*x)*A(x)^2 ) = x^3 + 3*x^4 - 30*x^5 + 70*x^6 + 195*x^7 - 2391*x^8 + 11467*x^9 - 30645*x^10 + 11340*x^11 + 354870*x^12 + ...

%e B(x)^5 = A( x^2*(1 + 5*x)*A(x)^3 ) = x^5 + 5*x^6 - 45*x^7 + 15*x^8 + 1110*x^9 - 6354*x^10 + 12315*x^11 + 64365*x^12 - 658755*x^13 + 2858405*x^14 + ...

%e B(x)^15 = A( x*(1 + 3*x)*A(x)^2 )^5 = x^15 + 15*x^16 - 60*x^17 - 1180*x^18 + 6480*x^19 + 41688*x^20 - 480825*x^21 + 497475*x^22 + 16467975*x^23 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); Ax = x*Ser(A);

%o A[#A] = polcoeff( subst(Ax,x, x^2*(1 + 5*x)*Ax^3 )^3 - subst(Ax,x, x*(1 + 3*x)*Ax^2 )^5, #A+14);); A[n]}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A370534, A370535 (dual), A370437, A370537.

%K sign

%O 1,3

%A _Paul D. Hanna_, Mar 08 2024