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A109084
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G.f. A(x) satisfies: A(x) = 1/G000041(x/A(x)) where G000041(x) is the g.f. of the partition numbers A000041.
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4
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1, -1, -2, -5, -17, -63, -253, -1062, -4615, -20570, -93538, -432211, -2023567, -9578815, -45767162, -220431025, -1069079067, -5216655257, -25592441875, -126157044454, -624560659184, -3103962569509, -15480272621533, -77450458331100, -388627340240958, -1955249529839424
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OFFSET
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0,3
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COMMENTS
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Note: coefficient [x^n] A(x)^n = -A000203(n) (sum of divisors of n) for n>0.
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LINKS
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FORMULA
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G.f.: A(x) = x/series_reversion(x*eta(x)). G.f.: A(x) = 1/G109085(x) where G109085(x) is g.f. of A109085.
a(n) ~ -c * d^n / n^(3/2), where d = A270915 = 5.35270133348664268777241581416... and c = 0.146705445870000769931272287955221766131167... - Vaclav Kotesovec, May 13 2018
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EXAMPLE
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The initial terms [x^0] through [x^n] of n-th self-convolution
are persistently small:
A^0: 1;
A^1: 1,-1;
A^2: 1,-2,-3;
A^3: 1,-3,-3,-4;
A^4: 1,-4,-2,0,-7;
A^5: 1,-5,0,5,0,-6;
A^6: 1,-6,3,10,3,6,-12;
A^7: 1,-7,7,14,0,7,0,-8;
A^8: 1,-8,12,16,-10,0,-8,8,-15;
A^9: 1,-9,18,15,-27,-9,-21,0,0,-13;
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MATHEMATICA
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(* Calculation of constant c: *) val = Sqrt[r*s^5*(-1 + s/r)*(Log[r/s]^2 / (2*Pi*(2*s^3*(-s*Log[1 - r/s] + ArcTanh[1 - 2*r/s] * (2*r - (r - s)*(Log[1 - r/s] - 2*Log[r/s]))) + (r - s)*(s^3*(2 - 2*Log[1 - r/s] + 3*Log[r/s]) * QPolyGamma[0, 1, r/s] - s^3*QPolyGamma[0, 1, r/s]^2 + s^3*QPolyGamma[1, 1, r/s] + r*Log[r/s]*(r*Log[r/s] * Derivative[0, 2][QPochhammer][r/s, r/s] - 2*s^2*Derivative[0, 0, 1][QPolyGamma][0, 1, r/s])))))] /. FindRoot[{QPochhammer[r/s] == s, (Log[1 - r/s] + QPolyGamma[0, 1, r/s])/Log[r/s] == 1 + (r*Derivative[0, 1][QPochhammer][r/s, r/s])/s^2}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 1000]; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Oct 02 2023 *)
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PROG
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(PARI) a(n)=polcoeff(x/serreverse(x*eta(x+x*O(x^n))), n)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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