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A370443
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Expansion of g.f. A(q) satisfying -3 = Product_{n>=0} (1 - 4*q^n*A(q)).
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5
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1, 3, 24, 216, 2184, 23592, 267144, 3128472, 37582680, 460564632, 5735093832, 72359126376, 923021734344, 11884281689688, 154243249784856, 2015831498613720, 26506024201097352, 350404606655241768, 4654501489433893512, 62092356103141330584, 831534637662059617368
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
(1) -3 = Product_{n>=0} (1 - 4*q^n*A(q)).
(2) -3 = Sum_{n>=0} (-4)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).
(3) A(q) = lim_{n->oo} R_{n}(q) / R_{n+1}(q) where R_{n}(q) = faq(n,q) * [x^n] 4/(1 + 3*e_q(4*x,q)), e_q(x,q) is the q-exponential of x and faq(n,q) is the q-factorial of n.
a(n) ~ c * d^n / n^(3/2), where d = 14.4231123176639449630408542507057843543532473958120624505916750124669... and c = 0.51707658317945675859732872615636765308571079799394608380170536716694... - Vaclav Kotesovec, Feb 18 2024
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EXAMPLE
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G.f.: A(q) = 1 + 3*q + 24*q^2 + 216*q^3 + 2184*q^4 + 23592*q^5 + 267144*q^6 + 3128472*q^7 + 37582680*q^8 + 460564632*q^9 + 5735093832*q^10 + ...
where A(q) satisfies the infinite product
-3 = (1 - 4*A(q)) * (1 - 4*q*A(q)) * (1 - 4*q^2*A(q)) * (1 - 4*q^3*A(q)) * (1 - 4*q^4*A(q)) * (1 - 4*q^5*A(q)) * ...
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MATHEMATICA
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(* Calculation of constants {d, c}: *) With[{m = 4}, {1/r, -s*Log[r] * Sqrt[r*Derivative[0, 1][QPochhammer][m*s, r] / (2*Pi*(m - 1)*QPolyGamma[1, Log[m*s]/Log[r], r])]} /. FindRoot[{m + QPochhammer[m*s, r] == 1, Log[1 - r] + QPolyGamma[0, Log[m*s]/Log[r], r] == 0}, {r, 1/m^2}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Feb 18 2024 *)
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PROG
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(PARI) /* A(q) satisfies -3 = Product_{n>=0} (1 - 4*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( 3 + prod(k=0, #A, 1 - 4*x^k*Ser(A)) /4, #A-1, x) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
{faq(n, q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
{R(n) = faq(n, q) * polcoeff( 4/(1 + 3*sum(m=0, n, (4*x)^m/faq(m, q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n+1)/R(n+2) + q*O(q^n), n, q)}
for(n=0, 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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