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A370442
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Expansion of g.f. A(q) satisfying -2 = Product_{n>=0} (1 - 3*q^n*A(q)).
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5
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1, 2, 12, 78, 570, 4434, 36174, 305142, 2640612, 23311068, 209111736, 1900666896, 17466522690, 162014855658, 1514885838582, 14263411673472, 135117683341050, 1286880634334490, 12315286940334942, 118362499698060384, 1141990331203349562, 11056838563337857548, 107394670044059002968
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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) -2 = Product_{n>=0} (1 - 3*q^n*A(q)).
(2) -2 = Sum_{n>=0} (-3)^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] 3/(1 + 2*e_q(3*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 = 10.3914965886269147720490605009350781702243358825286425537327254915874... and c = 0.49970395101356434785108820969954986510927554236884857759717688447784... - Vaclav Kotesovec, Feb 18 2024
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EXAMPLE
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G.f.: A(q) = 1 + 2*q + 12*q^2 + 78*q^3 + 570*q^4 + 4434*q^5 + 36174*q^6 + 305142*q^7 + 2640612*q^8 + 23311068*q^9 + 209111736*q^10 + ...
where A(q) satisfies the infinite product
-2 = (1 - 3*A(q)) * (1 - 3*q*A(q)) * (1 - 3*q^2*A(q)) * (1 - 3*q^3*A(q)) * (1 - 3*q^4*A(q)) * (1 - 3*q^5*A(q)) * ...
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MATHEMATICA
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(* Calculation of constants {d, c}: *) With[{m = 3}, {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 -2 = Product_{n>=0} (1 - 3*q^n*A(q)) */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( 2 + prod(k=0, #A, 1 - 3*x^k*Ser(A)) /3, #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( 3/(1 + 2*sum(m=0, n, (3*x)^m/faq(m, q) + x*O(x^(n+2)))), n, x)}
{a(n) = polcoeff(R(n+2)/R(n+3) + 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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