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%I #9 Oct 02 2023 13:28:29
%S 1,-1,-2,-5,-17,-63,-253,-1062,-4615,-20570,-93538,-432211,-2023567,
%T -9578815,-45767162,-220431025,-1069079067,-5216655257,-25592441875,
%U -126157044454,-624560659184,-3103962569509,-15480272621533,-77450458331100,-388627340240958,-1955249529839424
%N G.f. A(x) satisfies: A(x) = 1/G000041(x/A(x)) where G000041(x) is the g.f. of the partition numbers A000041.
%C Note: coefficient [x^n] A(x)^n = -A000203(n) (sum of divisors of n) for n>0.
%H Vaclav Kotesovec, <a href="/A109084/b109084.txt">Table of n, a(n) for n = 0..1000</a>
%F 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.
%F a(n) ~ -c * d^n / n^(3/2), where d = A270915 = 5.35270133348664268777241581416... and c = 0.146705445870000769931272287955221766131167... - _Vaclav Kotesovec_, May 13 2018
%e The initial terms [x^0] through [x^n] of n-th self-convolution
%e are persistently small:
%e A^0: 1;
%e A^1: 1,-1;
%e A^2: 1,-2,-3;
%e A^3: 1,-3,-3,-4;
%e A^4: 1,-4,-2,0,-7;
%e A^5: 1,-5,0,5,0,-6;
%e A^6: 1,-6,3,10,3,6,-12;
%e A^7: 1,-7,7,14,0,7,0,-8;
%e A^8: 1,-8,12,16,-10,0,-8,8,-15;
%e A^9: 1,-9,18,15,-27,-9,-21,0,0,-13;
%t (* 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 *)
%o (PARI) a(n)=polcoeff(x/serreverse(x*eta(x+x*O(x^n))),n)
%Y Cf. A109085, A000041, A000203.
%K sign
%O 0,3
%A _Paul D. Hanna_, Jun 18 2005
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