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%I #9 Jan 17 2024 10:10:41
%S 1,1,4,14,59,258,1187,5623,27302,135063,678468,3451272,17742514,
%T 92034588,481112574,2532013892,13404579322,71336740012,381416582710,
%U 2047875323729,11036900422910,59686672359369,323788693045886,1761507417706018
%N G.f. A(x) satisfies: 1 + x = P(A(x)) / Q(A(x)), where P(x) = Product_{n>=0} (1 + x^(5*n+1))*(1 + x^(5*n+4)) and Q(x) = Product_{n>=0} (1 + x^(5*n+2))*(1 + x^(5*n+3)), with A(0) = 0, A'(0) = 1.
%H Vaclav Kotesovec, <a href="/A351634/b351634.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: A(x) = Series_Reversion( P(x)/Q(x) - 1 ), where P(x) = Product_{n>=0} (1 + x^(5*n+1))*(1 + x^(5*n+4)) and Q(x) = Product_{n>=0} (1 + x^(5*n+2))*(1 + x^(5*n+3)), with A(0) = 0, A'(0) = 1.
%F a(n) ~ c * d^n / n^(3/2), where d = 5.7997668905429653202956499676894864614337725024680731963895428378920947... and c = 0.0983896146762908218558422941662822756464709531976861748855671955... - _Vaclav Kotesovec_, Mar 15 2022
%e G.f.: A(x) = x + x^2 + 4*x^3 + 14*x^4 + 59*x^5 + 258*x^6 + 1187*x^7 + 5623*x^8 + 27302*x^9 + 135063*x^10 + 678468*x^11 + 3451272*x^12 + ...
%e where A = A(x) satisfies the infinite product:
%e 1 + x = (1 + A)*(1 + A^4)/((1 + A^2)*(1 + A^3)) * (1 + A^6)*(1 + A^9)/((1 + A^7)*(1 + A^8)) * (1 + A^11)*(1 + A^14)/((1 + A^12)*(1 + A^13)) * (1 + A^16)*(1 + A^19)/((1 + A^17)*(1 + A^18)) * ...
%e equivalently,
%e 1 + x = P(A(x)) / Q(A(x))
%e where
%e P(A(x)) = 1 + x + x^2 + 4*x^3 + 15*x^4 + 64*x^5 + 286*x^6 + 1332*x^7 + 6378*x^8 + 31224*x^9 + 155527*x^10 + ...
%e Q(A(x)) = 1 + x^2 + 3*x^3 + 12*x^4 + 52*x^5 + 234*x^6 + 1098*x^7 + 5280*x^8 + 25944*x^9 + 129583*x^10 + ...
%e and
%e P(x) = 1 + x + x^4 + x^5 + x^6 + x^7 + x^9 + 2*x^10 + 2*x^11 + x^12 + x^13 + 2*x^14 + ... + A203776(n)*x^n + ...
%e Q(x) = 1 + x^2 + x^3 + x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + 2*x^12 + 2*x^13 + x^14 + ... + A219607(n)*x^n + ...
%e also
%e P(x)/Q(x) = 1 + x - x^2 - 2*x^3 + x^4 + 3*x^5 + x^6 - 3*x^7 - 4*x^8 + x^9 + 6*x^10 + 3*x^11 - 6*x^12 - 8*x^13 + 2*x^14 + 12*x^15 + 6*x^16 - 11*x^17 - 15*x^18 + 3*x^19 + 22*x^20 + ...
%e such that A(x) = Series_Reversion(P(x)/Q(x) - 1).
%t Rest[CoefficientList[InverseSeries[Series[-1 + QPochhammer[-x, x^5] * QPochhammer[-x^4, x^5] / (QPochhammer[-x^2, x^5] * QPochhammer[-x^3, x^5]), {x, 0, 25}], x], x]] (* _Vaclav Kotesovec_, Jan 17 2024 *)
%t (* Calculation of constant d: *) 1/r /. FindRoot[{1 + r == QPochhammer[-s, s^5]* QPochhammer[-s^4, s^5]/(QPochhammer[-s^2, s^5] * QPochhammer[-s^3, s^5]), 5*s^4*QPochhammer[-s^4, s^5]* Derivative[0, 1][QPochhammer][-s, s^5] + 1/s*QPochhammer[-s, s^5] * (-1/Log[s^5] * QPochhammer[-s^4, s^5]*(QPolyGamma[0, Log[-s]/Log[s^5], s^5] - 2*QPolyGamma[0, Log[-s^2]/Log[s^5], s^5] - 3*QPolyGamma[0, Log[-s^3]/Log[s^5], s^5] + 4*QPolyGamma[0, Log[-s^4]/Log[s^5], s^5]) - 5*s^5*QPochhammer[-s^4, s^5]*Derivative[0, 1][QPochhammer][ -s^2, s^5]/QPochhammer[-s^2, s^5] - 5*s^5*QPochhammer[-s^4, s^5] * Derivative[0, 1][QPochhammer][-s^3, s^5]/QPochhammer[-s^3, s^5] + 5*s^5*Derivative[0, 1][QPochhammer][-s^4, s^5]) == 0}, {r, 1/5}, {s, 1/3}, WorkingPrecision -> 80] (* _Vaclav Kotesovec_, Jan 17 2024 *)
%o (PARI) /* As Series Reversion of P(x)/Q(x) - 1 */
%o {a(n) = my(A=x,P,Q);
%o P = prod(m=0,n, (1 + x^(5*m+1))*(1 + x^(5*m+4)) +x*O(x^n));
%o Q = prod(m=0,n, (1 + x^(5*m+2))*(1 + x^(5*m+3)) +x*O(x^n));
%o polcoeff( serreverse( P/Q - 1 ), n)}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) /* Obtain A(x) Using P(A(x))/Q(A(x)) = 1 + x */
%o {a(n) = my(A=[0,1],P,Q); for(i=1,n, A = concat(A,0);
%o PA = prod(m=0,#A, (1 + Ser(A)^(5*m+1))*(1 + Ser(A)^(5*m+4)) );
%o QA = prod(m=0,#A, (1 + Ser(A)^(5*m+2))*(1 + Ser(A)^(5*m+3)) );
%o A[#A] = -polcoeff( PA/QA ,#A-1) );A[n+1]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A203776 (P), A219607 (Q).
%K nonn
%O 1,3
%A _Paul D. Hanna_, Mar 14 2022
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