Original entry on oeis.org
1, 8, 50, 240, 1024, 3888, 13696, 44960, 139970, 414904, 1181568, 3242928, 8623104, 22268752, 56039936, 137686048, 331039232, 780029536, 1804321074, 4102056144, 9177497600, 20225408480, 43948974720, 94236510112, 199549448704
Offset: 0
G.f.: B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...
Bisection B_1(q) of A162584 begins:
B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...
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eta[q_]:= q^(1/24)*QPochhammer[q]; nmax =250; a[n_]:= SeriesCoefficient[ Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], 2*n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
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{a(n)=local(L=sum(m=1, 2*n, 2*sigma(m)*2^valuation(m, 2)*x^m/m)+O(x^(2*n+1))); polcoeff(exp(L), 2*n)}
Original entry on oeis.org
2, 16, 96, 448, 1858, 6896, 23776, 76608, 234432, 684336, 1921472, 5206208, 13679490, 34941120, 87036576, 211822976, 504784704, 1179589728, 2707337056, 6109982400, 13575320320, 29721857904, 64184237216, 136816242816
Offset: 1
G.f.: B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...
Bisection B_0(q) of A162584 begins:
B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...
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eta[q_]:= q^(1/24)*QPochhammer[q]; nmax =250; a[n_]:= SeriesCoefficient[ Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], 2*n + 1]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
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{a(n)=local(L=sum(m=1, 2*n+1, 2*sigma(m)*2^valuation(m, 2)*x^m/m)+O(x^(2*n+2))); polcoeff(exp(L), 2*n+1)}
A163129
G.f.: A(q) = exp( Sum_{n>=1} sigma(n) * 3*A038500(n) * q^n/n ), where A038500(n) = highest power of 3 dividing n.
Original entry on oeis.org
1, 3, 9, 30, 75, 180, 441, 969, 2070, 4431, 8964, 17775, 35094, 66975, 125865, 235053, 429096, 773766, 1386027, 2442372, 4260645, 7384578, 12640320, 21453975, 36192519, 60454713, 100250100, 165311094, 270391857, 439479198, 710631279
Offset: 0
G.f.: A(q) = 1 + 3*q + 9*q^2 + 30*q^3 + 75*q^4 + 180*q^5 + 441*q^6 + ...
log(A(q)) = 3*q + 9*q^2/2 + 36*q^3/3 + 21*q^4/4 + 18*q^5/5 + 108*q^6/6 + ...
Define TRISECTIONS:
T_0(q) = 1 + 30*q^3 + 441*q^6 + 4431*q^9 + 35094*q^12 + ...
T_1(q) = 3*q + 75*q^4 + 969*q^7 + 8964*q^10 + 66975*q^13 + ...
T_2(q) = 9*q^2 + 180*q^5 + 2070*q^8 + 17775*q^11 + 125865*q^14 + ...
then:
3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091:
T9B(q) = 1/q + 5*q^2 - 7*q^5 + 3*q^8 + 15*q^11 - 32*q^14 + 9*q^17 + 58*q^20 + ...
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nmax = 100; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*3^(IntegerExponent[k, 3] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], q] (* G. C. Greubel, Jul 03 2018, edited by Vaclav Kotesovec, Oct 20 2020 *)
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{a(n)=local(L=sum(m=1, n, 3*sigma(m)*3^valuation(m, 3)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}
Showing 1-3 of 3 results.