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 1-4 of 4 results.

A163130 A trisection of A163129.

Original entry on oeis.org

1, 30, 441, 4431, 35094, 235053, 1386027, 7384578, 36192519, 165311094, 710631279, 2897149824, 11270295093, 42043460145, 151025654781, 524199355128, 1763256696537, 5762466306432, 18337081016448, 56926806819666
Offset: 0

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_0(q) = 1 + 30*q^3 + 441*q^6 + 4431*q^9 + 35094*q^12 + ...

Links

Crossrefs

Cf. A163129, A163131 (T_1), A163132 (T_2), A058091.

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]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
  • PARI
    {a(n)=local(L=sum(m=1, 3*n, 3*sigma(m)*3^valuation(m, 3)*x^m/m)+x*O(x^(3*n))); polcoeff(exp(L), 3*n)}

Extensions

Comment corrected by Paul D. Hanna, Jul 24 2009

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

A162584 G.f.: A(x) = exp( 2*Sum_{n>=1} sigma(n)*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 8, 16, 50, 96, 240, 448, 1024, 1858, 3888, 6896, 13696, 23776, 44960, 76608, 139970, 234432, 414904, 684336, 1181568, 1921472, 3242928, 5206208, 8623104, 13679490, 22268752, 34941120, 56039936, 87036576, 137686048, 211822976
Offset: 0

Views

Author

Paul D. Hanna, Jul 06 2009

Keywords

Comments

Log of the g.f. A(x) is formed from the term-wise product of the log of the g.f.s of the partition numbers A000041 and the binary partitions A000123.

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 16*x^3 + 50*x^4 + 96*x^5 + 240*x^6 + ...
log(A(x))/2 = x + 6*x^2/2 + 4*x^3/3 + 28*x^4/4 + 6*x^5/5 + 24*x^6/6 + 8*x^7/7 + 120*x^8/8 + ... + sigma(n)*A006519(n)*x^n/n + ...
The log of the g.f. of the Partition numbers (A000041) is:
x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 + 12*x^6/6 + ... + sigma(n)*x^n/n + ...
The log of the g.f. of the binary partitions (A000123) is:
x + x^2/2 + x^3/3 + 4*x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 + ... + A006519(n)*x^n/n + ...
From _Paul D. Hanna_, Jul 26 2009: (Start)
BISECTIONS begin:
B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...
B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...
where 2*B_0(q)/B_1(q) = T16B(q):
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...
which is a g.f. of A029839. (End)

Links

Crossrefs

Cf. A000203, A006519, A000041, A000123.
Cf. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. - Paul D. Hanna, Jul 26 2009

Programs

  • Mathematica
    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}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 20 2020 *)
nmax = 40; CoefficientList[Series[Product[1/EllipticTheta[4, 0, x^(2^k)]^(2^k), {k, 0, 1 + Log[2, nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)
  • PARI
    {a(n)=local(L=sum(m=1,n,2*sigma(m)*2^valuation(m,2)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

Formula

From Paul D. Hanna, Jul 26 2009: (Start)
Define series BISECTIONS A(q) = B_0(q) + B_1(q), then
2*B_0(q)/B_1(q) = T16B(q) = q*eta(q^8)^6/(eta(q^4)^2*eta(q^16)^4), the McKay-Thompson series of class 16B for the Monster group (A029839). (End)
G.f.: 1/Product_{n>=0} Theta4(q^(2^n))^(2^n) = 1 / ( E(1)^2*E(2)^3*E(4)^6*E(8)^12* ... * E(2^n)^A042950(n) * ... ) where E(n) = Product_{k>=1} (1-q^(n*k)). - Joerg Arndt, Mar 20 2010
Compare to the previous formula: 1/Product_{n>=0} Theta3(q^(2^n))^(2^n) = Theta4(q). - Joerg Arndt, Aug 03 2011
Showing 1-4 of 4 results.

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