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%I #10 Feb 18 2024 14:58:47
%S 1,5,60,840,13200,222060,3914880,71382000,1335069060,25471344000,
%T 493782003960,9698680513380,192596799930360,3860313616039080,
%U 77994531433117500,1586781765309289080,32479537197876474480,668397905439059302860,13820909295920888641200,287010766542642877455600
%N Expansion of g.f. A(q) satisfying -5 = Product_{n>=0} (1 - 6*q^n*A(q)).
%H Paul D. Hanna, <a href="/A370445/b370445.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(q) = Sum_{n>=0} a(n)*q^n satisfies the following formulas.
%F (1) -5 = Product_{n>=0} (1 - 6*q^n*A(q)).
%F (2) -5 = Sum_{n>=0} (-6)^n * q^(n*(n-1)/2) * A(q)^n / Product_{k=1..n} (1 - q^k).
%F (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.
%F a(n) ~ c * d^n / n^(3/2), where d = 22.4513560224097075964072006016077322187366268210808011681398076264064... and c = 0.53352515179176712675216292204161595268714373520357919375218426836012... - _Vaclav Kotesovec_, Feb 18 2024
%e 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 + ...
%e where A(q) satisfies the infinite product
%e -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)) * ...
%t (* 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 *)
%o (PARI) /* A(q) satisfies -5 = Product_{n>=0} (1 - 6*q^n*A(q)) */
%o {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A[#A] = polcoeff( 5 + prod(k=0,#A, 1 - 6*x^k*Ser(A)) /6, #A-1, x) ); H=A; A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* limit_{n->oo} R_{n}(q) / R_{n+1}(q) */
%o {faq(n,q) = prod(j=1, n, (q^j-1)/(q-1))} \\ q-factorial of n
%o {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)}
%o {a(n) = polcoeff(R(n+1)/R(n+2) + q*O(q^n), n, q)}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A152807, A370442, A370443, A370444.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 18 2024
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