cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Paul D. Hanna, Jul 21 2009, Jul 24 2009

Keywords

Examples

			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 + ...

Links

Crossrefs

Cf. trisections: A163130 (T_0), A163131 (T_1), A163132 (T_2).
Cf. A058091, A038500, A162584 (variant).

Programs

  • Mathematica
    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 *)
  • PARI
    {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)}

Formula

Define trisections by: A(q) = T_0(q) + T_1(q) + T_2(q), then:
3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091,
which is the McKay-Thompson series of class 9B for Monster.
G.f.: 1/Product_{n>=0} R(q^(3^n))^(3^n) where R(q) = E(q)^3/E(q^3) and E(q) = Product_{k>=1} (1 - q^k). - Joerg Arndt, Aug 03 2011

A163131 A trisection of A163129.

Original entry on oeis.org

3, 75, 969, 8964, 66975, 429096, 2442372, 12640320, 60454713, 270391857, 1141260315, 4578160257, 17554638039, 64642406670, 229486544439, 788018124312, 2624648438025, 8499852952224, 26820711864657, 82613109082410
Offset: 1

Views

Author

Paul D. Hanna, Jul 21 2009

Keywords

Comments

A163129 is defined by the 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.
Trisections are related by: A(q) = T_0(q) + T_1(q) + T_2(q) where
3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091,
which is the McKay-Thompson series of class 9B for Monster.

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • 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)}

Extensions

Comment corrected by Paul D. Hanna, Jul 24 2009

A163132 A trisection of A163129.

Original entry on oeis.org

9, 180, 2070, 17775, 125865, 773766, 4260645, 21453975, 100250100, 439479198, 1822654251, 7198716870, 27221451885, 98988000120, 347428124352, 1180620288702, 3894719205510, 12501561121560, 39124469772495
Offset: 2

Views

Author

Paul D. Hanna, Jul 21 2009

Keywords

Comments

A163129 is defined by the 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.
Trisections are related by: A(q) = T_0(q) + T_1(q) + T_2(q) where
3*T_0(q)/T_1(q) = 3*T_1(q)/T_2(q) = T9B(q), the g.f. of A058091,
which is the McKay-Thompson series of class 9B for Monster.

Examples

			G.f.: T_2(q) = 9*q^2 + 180*q^5 + 2070*q^8 + 17775*q^11 + 125865*q^14 + ...
Terms are divisible by 9:
T_2/9 = [1, 20, 230, 1975, 13985, 85974, 473405, 2383775, 11138900, ...].

Links

Crossrefs

Cf. A163129, A163130 (T_0), A163131 (T_1), A058091, A038500.

Programs

  • Mathematica
    eta[q_] := q^(1/24)*QPochhammer[q]; nmax = 250; 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 + 2]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
  • PARI
    {a(n)=local(L=sum(m=1, 3*n+2, 3*sigma(m)*3^valuation(m, 3)*x^m/m)+x*O(x^(3*n+2))); polcoeff(exp(L), 3*n+2)}

Extensions

Comment corrected by Paul D. Hanna, Jul 24 2009
Showing 1-3 of 3 results.

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