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.

A308290 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k)^3)/k.

Original entry on oeis.org

1, 2, 3, 1, -1, -6, -3, 2, 9, 9, -6, -24, -25, 16, 72, 75, -35, -213, -239, 78, 627, 767, -182, -1890, -2477, 355, 5847, 8109, -360, -18195, -26801, -1225, 56724, 89040, 11431, -177897, -297030, -61857, 560310, 994427, 284075, -1766754, -3338212, -1201932
Offset: 1

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Author

Ilya Gutkovskiy, May 18 2019

Keywords

Comments

Inverse Euler transform of triangular numbers (A000217).

Crossrefs

Cf. A000217, A000294, A008683, A308291, A316152.

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/(1 - x^k)^3]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j^2*(j + 1)/2 - Sum[s[[d]]*(j - d)*(j - d + 1)/2, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 10 2019 *)

Formula

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000217.

A308297 Expansion of Sum_{k>=1} mu(k)*log((theta_3(x^k) + 1)/2)/k, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 2, -3, 1, 1, -2, 5, -6, 4, -2, -3, 10, -15, 15, -9, -1, 17, -34, 43, -39, 17, 25, -78, 117, -127, 93, 3, -147, 298, -394, 369, -168, -211, 680, -1092, 1251, -939, 39, 1336, -2827, 3855, -3715, 1857, 1777, -6529, 10922, -12789, 9929
Offset: 1

Views

Author

Ilya Gutkovskiy, May 19 2019

Keywords

Comments

Inverse Euler transform of A010052.

Crossrefs

Cf. A000122, A001156, A008683, A010052, A316152.

Programs

  • Mathematica
    nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k] Log[(EllipticTheta[3, 0, x^k] + 1)/2]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 57; CoefficientList[Series[Sum[MoebiusMu[k] Log[Sum[x^(j^2 k), {j, 0, nmax}]]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} mu(k)*log(Sum_{j>=0} x^(j^2*k))/k.
Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A010052.

A316150 Inverse Weigh transform of n^2.

Original entry on oeis.org

1, 4, 5, 5, -6, -12, -4, 34, 56, 1, -158, -267, 56, 874, 1234, -691, -4966, -5796, 6132, 28411, 26932, -46687, -162814, -117746, 332350, 929348, 454328, -2278344, -5259270, -1250947, 15199212, 29375294, -1279006, -99217863, -161079712, 60427836, 635914664
Offset: 1

Views

Author

Seiichi Manyama, Jun 25 2018

Keywords

Examples

			(1+x)*(1+x^2)^4*(1+x^3)^5*(1+x^4)^5*(1+x^5)^(-6)* ... = 1 + x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + ... .

Links

Crossrefs

Cf. A000290, A253909, A306152, A316152.

Formula

Product_{k>=1} (1+x^k)^a(k) = 1 + Sum_{k>=1} A000290(k)*x^k.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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