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A355361
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G.f. A(x) satisfies: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
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11
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1, 1, 5, 26, 136, 746, 4261, 25173, 152596, 943804, 5931561, 37768700, 243124702, 1579577423, 10344340396, 68212177180, 452531832109, 3018280278965, 20227324602249, 136135295125566, 919757424512780, 6235752585125348, 42411283395662960, 289289349007740037
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OFFSET
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0,3
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COMMENTS
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Equals the row sums of triangle A355360: a(n) = Sum_{k=0..n} A355360(n,k) for n >= 0.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 7.27539777267340429262199058476266... and c = 0.504162727798216251681995853318... - Vaclav Kotesovec, Jul 20 2022
A(1/d) = 1.78721033673795569... where 1/d = 0.1374495293928845... and d is the value given above by Vaclav Kotesovec. - Paul D. Hanna, Jul 30 2022
Formula (3) can be rewritten as the functional equation QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)) = x*y / QPochhammer(x). - Vaclav Kotesovec, Jan 19 2024
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EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 136*x^4 + 746*x^5 + 4261*x^6 + 25173*x^7 + 152596*x^8 + 943804*x^9 + 5931561*x^10 + ...
where
x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...
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MATHEMATICA
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(* Calculation of constants {d, c}: *) {1/r, s*Sqrt[((-1 + s)*(-1 + r*s) * Log[r]*((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r] - r*(-1 + s)*(-1 + r*s) * Log[r] * Derivative[0, 1][QPochhammer][r, r] / QPochhammer[r] + r*Log[r]*QPochhammer[r] * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + (-1 + r*s)*((1 - s) * QPolyGamma[0, Log[s]/Log[r], r] - Log[r]*(s + r*(-1 + s) * Derivative[0, 1][QPochhammer][s, r] / QPochhammer[s, r])))) / (2* Pi*(-s*(1 + r - 4*r*s + r*(1 + r)*s^2) * Log[r]^2 + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, -Log[r*s]/Log[r], r]))]} /. FindRoot[{QPochhammer[r] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] / ((-1 + s)*(-1 + r*s)) == -1, 1/(-1 + s) + 1/(-1 + r*s) + (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/(r*s)]/Log[r], r])/Log[r] == -2}, {r, 1/7}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
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PROG
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(PARI) {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A) - sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
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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