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A351634
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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.
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1
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1, 1, 4, 14, 59, 258, 1187, 5623, 27302, 135063, 678468, 3451272, 17742514, 92034588, 481112574, 2532013892, 13404579322, 71336740012, 381416582710, 2047875323729, 11036900422910, 59686672359369, 323788693045886, 1761507417706018
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OFFSET
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1,3
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LINKS
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FORMULA
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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.
a(n) ~ c * d^n / n^(3/2), where d = 5.7997668905429653202956499676894864614337725024680731963895428378920947... and c = 0.0983896146762908218558422941662822756464709531976861748855671955... - Vaclav Kotesovec, Mar 15 2022
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EXAMPLE
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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 + ...
where A = A(x) satisfies the infinite product:
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)) * ...
equivalently,
1 + x = P(A(x)) / Q(A(x))
where
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 + ...
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 + ...
and
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 + ...
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 + ...
also
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 + ...
such that A(x) = Series_Reversion(P(x)/Q(x) - 1).
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MATHEMATICA
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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 *)
(* 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 *)
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PROG
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(PARI) /* As Series Reversion of P(x)/Q(x) - 1 */
{a(n) = my(A=x, P, Q);
P = prod(m=0, n, (1 + x^(5*m+1))*(1 + x^(5*m+4)) +x*O(x^n));
Q = prod(m=0, n, (1 + x^(5*m+2))*(1 + x^(5*m+3)) +x*O(x^n));
polcoeff( serreverse( P/Q - 1 ), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Obtain A(x) Using P(A(x))/Q(A(x)) = 1 + x */
{a(n) = my(A=[0, 1], P, Q); for(i=1, n, A = concat(A, 0);
PA = prod(m=0, #A, (1 + Ser(A)^(5*m+1))*(1 + Ser(A)^(5*m+4)) );
QA = prod(m=0, #A, (1 + Ser(A)^(5*m+2))*(1 + Ser(A)^(5*m+3)) );
A[#A] = -polcoeff( PA/QA , #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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