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A171805
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G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).
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4
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1, 3, 18, 130, 1044, 8946, 80135, 741312, 7027515, 67911855, 666525630, 6625647054, 66570488901, 674964968175, 6897258376218, 70961851119848, 734455079297433, 7641851681095236, 79886815507105175, 838655487787502616, 8837797224686207976, 93454820274339167191
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;
(2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).
G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
Self-convolution cube of A171804 (with offset).
a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - Vaclav Kotesovec, Nov 11 2017
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EXAMPLE
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G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...
where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and
x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...
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MATHEMATICA
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(* Calculation of constants {d, c}: *) eq = FindRoot[{r/QPochhammer[s]^3 == s, 1/s + 3*(s/r)^(1/3)*Derivative[0, 1][QPochhammer][s, s] == (3*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/10}, {s, 1/8}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*(-1 + s)*s^2*(Log[s]^2/(6*Pi*(r*(-4*s*ArcTanh[1 - 2*s] + Log[1 - s]*(2 + 3*(-1 + s)*Log[1 - s] + Log[s] - s*Log[s])) - (-1 + s)*(-3*r*QPolyGamma[0, 1, s]^2 + r*QPolyGamma[1, 1, s] + QPolyGamma[0, 1, s]*(r*(2 - 6*Log[1 - s] + Log[s]) + 6*(r/s)^(2/3)*s^2*Log[s]* Derivative[0, 1][QPochhammer][s, s]) + s*Log[s]*((r/s)^(1/3)*s*(6*(r/s)^(1/3) * Log[1 - s] * Derivative[0, 1][QPochhammer][s, s] - 4*s*Log[s] * Derivative[0, 1][QPochhammer][s, s]^2 + (r/s)^(1/3)*s*Log[s]* Derivative[0, 2][QPochhammer][s, s]) - 2*r*Derivative[0, 0, 1][ QPolyGamma][0, 1, s])))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)
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PROG
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(PARI) {a(n)=polcoeff(serreverse(x*eta(x+x*O(x^n))^3), 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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EXTENSIONS
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STATUS
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approved
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