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A trisection of A163129.
4

%I #7 Jul 04 2018 08:58:00

%S 3,75,969,8964,66975,429096,2442372,12640320,60454713,270391857,

%T 1141260315,4578160257,17554638039,64642406670,229486544439,

%U 788018124312,2624648438025,8499852952224,26820711864657,82613109082410

%N A trisection of A163129.

%C A163129 is defined by the g.f.:

%C A(q) = exp( Sum_{n>=1} sigma(n) * 3*A038500(n) * q^n/n ),

%C where A038500(n) = highest power of 3 dividing n.

%C Trisections are related by: A(q) = T_0(q) + T_1(q) + T_2(q) where

%C 3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091,

%C which is the McKay-Thompson series of class 9B for Monster.

%H G. C. Greubel, <a href="/A163131/b163131.txt">Table of n, a(n) for n = 1..1001</a>

%e G.f.: T_1(q) = 3*q + 75*q^4 + 969*q^7 + 8964*q^10 + 66975*q^13 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 150; a[n_]:= SeriesCoefficient[Series[Exp[Sum[DivisorSigma[1, k]* 3^(IntegerExponent[k, 3] + 1)*q^k/k, {k, 1, 3*nmax + 1}]], {q, 0, nmax}], 3*n + 1]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jul 03 2018 *)

%o (PARI) {a(n)=local(L=sum(m=1, 3*n+1, 3*sigma(m)*3^valuation(m, 3)*x^m/m)+x*O(x^(3*n+1))); polcoeff(exp(L), 3*n+1)}

%Y Cf. A163129, A163130 (T_0), A163132 (T_2), A058091, A038500.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jul 21 2009

%E Comment corrected by _Paul D. Hanna_, Jul 24 2009