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A370445
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Expansion of g.f. A(q) satisfying -5 = Product_{n>=0} (1 - 6*q^n*A(q)).
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5
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1, 5, 60, 840, 13200, 222060, 3914880, 71382000, 1335069060, 25471344000, 493782003960, 9698680513380, 192596799930360, 3860313616039080, 77994531433117500, 1586781765309289080, 32479537197876474480, 668397905439059302860, 13820909295920888641200, 287010766542642877455600
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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) -5 = Product_{n>=0} (1 - 6*q^n*A(q)).
(2) -5 = Sum_{n>=0} (-6)^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] 6/(1 + 5*e_q(6*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 = 22.4513560224097075964072006016077322187366268210808011681398076264064... and c = 0.53352515179176712675216292204161595268714373520357919375218426836012... - Vaclav Kotesovec, Feb 18 2024
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EXAMPLE
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G.f.: A(q) = 1 + 5*q + 60*q^2 + 840*q^3 + 13200*q^4 + 222060*q^5 + 3914880*q^6 + 71382000*q^7 + 1335069060*q^8 + 25471344000*q^9 + 493782003960*q^10 + ...
where A(q) satisfies the infinite product
-5 = (1 - 6*A(q)) * (1 - 6*q*A(q)) * (1 - 6*q^2*A(q)) * (1 - 6*q^3*A(q)) * (1 - 6*q^4*A(q)) * (1 - 6*q^5*A(q)) * ...
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MATHEMATICA
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(* Calculation of constants {d, c}: *) With[{m = 6}, {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 -5 = Product_{n>=0} (1 - 6*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( 5 + prod(k=0, #A, 1 - 6*x^k*Ser(A)) /6, #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( 6/(1 + 5*sum(m=0, n, (6*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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