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%I #13 Mar 08 2024 16:30:00
%S 1,0,-3,10,-15,84,-161,-174,612,1596,1926,-38592,-5895,234684,603621,
%T -2907414,-7559802,15762420,116671738,-95405184,-1326191061,
%U -582470792,14912006205,21762765822,-142157118077,-418666482912,1222704927801,6161408393758,-7256289555369,-80123028681972
%N Expansion of g.f. A(x) satisfying A( x*(1 + 2*x)*A(x) )^3 = A( x^2*(1 + 3*x)*A(x) )^2.
%H Paul D. Hanna, <a href="/A370437/b370437.txt">Table of n, a(n) for n = 1..501</a>
%e G.f.: A(X) = x - 3*x^3 + 10*x^4 - 15*x^5 + 84*x^6 - 161*x^7 - 174*x^8 + 612*x^9 + 1596*x^10 + 1926*x^11 - 38592*x^12 - 5895*x^13 + 234684*x^14 + 603621*x^15 + ...
%e where A( x*(1 + 2*x)*A(x) )^3 = A( x^2*(1 + 3*x)*A(x) )^2.
%e RELATED SERIES.
%e B(x) = A( x*(1 + 2*x)*A(x) )^(1/2) = A( x^2*(1 + 3*x)*A(x) )^(1/3)
%e where B(x) is the g.f. of A370438, which begins
%e B(x) = x + x^2 - 2*x^3 + 4*x^4 - 5*x^5 + 31*x^6 - 45*x^7 - 57*x^8 - 66*x^9 + 1124*x^10 + 116*x^11 - 8314*x^12 - 21328*x^13 + 76424*x^14 + 229013*x^15 + ...
%e B(x)^2 = A( x*(1 + 2*x)*A(x) ) = x^2 + 2*x^3 - 3*x^4 + 4*x^5 + 2*x^6 + 36*x^7 + 8*x^8 - 368*x^9 + 207*x^10 + 1674*x^11 + 3699*x^12 - 23640*x^13 - 51605*x^14 + 134174*x^15 + ...
%e B(x)^3 = A( x^2*(1 + 3*x)*A(x) ) = x^3 + 3*x^4 - 3*x^5 + x^6 + 15*x^7 + 39*x^8 + 88*x^9 - 684*x^10 + 36*x^11 + 3514*x^12 + 6807*x^13 - 33825*x^14 - 124742*x^15 + 217842*x^16 + ...
%e B(x)^6 = A( x*(1 + 2*x)*A(x) )^3 = x^6 + 6*x^7 + 3*x^8 - 16*x^9 + 45*x^10 + 162*x^11 + 321*x^12 - 1044*x^13 - 4257*x^14 + 12694*x^15 + 37275*x^16 - 61476*x^17 - 530776*x^18 + ...
%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 + 3*x)*Ax )^2 - subst(Ax,x, x*(1 + 2*x)*Ax )^3, #A+5);); A[n]}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A370438, A370537 (dual), A370535.
%K sign
%O 1,3
%A _Paul D. Hanna_, Mar 07 2024
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