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.

A308397 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(1 - x^(k^2))/k^2).

Original entry on oeis.org

1, 1, -1, -5, 7, 71, -59, -1511, -9295, -1583, 861751, 4039091, -80670281, -606807785, 7674244397, 78614840641, 1146707474401, 12874145737889, -1054507266321425, -19048413877999253, 238097060642380391, 6646823785301856871, -59731575523361439851, -2231444370433747995415
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Crossrefs

Cf. A008836, A118207, A190585, A205801, A306831, A308396, A308398.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (1 - x^(k^2))/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[(1 + x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(lambda(k)/k), where lambda() is the Liouville function (A008836).

A308396 Expansion of e.g.f. exp(-Sum_{k>=1} x^(k^2)/k^2).

Original entry on oeis.org

1, -1, 1, -1, -5, 29, -89, 209, 841, -50905, 458641, -2423521, 8243731, 158742869, -2450634185, 18519809489, -1402926535919, 21355930009679, -139305034406879, 306503668195775, 40578438892908331, -816475138658703091, 6941097158619626311, -24787202385366731311
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Crossrefs

Cf. A008836, A118205, A190585, A205801, A306831, A308397, A308398.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[-Sum[x^(k^2)/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: Product_{k>=1} (1 - x^k)^(lambda(k)/k), where lambda() is the Liouville function (A008836).

A308398 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(x^(k^2) - 1)/k^2).

Original entry on oeis.org

1, -1, 3, -7, 19, -51, 61, 167, 6777, -107929, 1650691, -17839911, 157217083, -1229269627, 6185945949, -3251776921, -1151787785999, 10138302541647, 532690324952707, -14122245788830279, 443912721023736291, -7480012715591067331, 115775303074594208893, -1392396864130912381017
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Crossrefs

Cf. A008836, A118208, A190585, A205801, A306831, A308396, A308397.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (x^(k^2) - 1)/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[1/(1 + x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: Product_{k>=1} 1/(1 + x^k)^(lambda(k)/k), where lambda() is the Liouville function (A008836).

A308381 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2)*(2 + x^(k^2))/(2*k^2)).

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 119296, 1075456, 9088256, 85292032, 894690304, 8968964096, 90882789376, 2409397682176, 40515889528832, 1051789297844224, 16251803853193216, 302342408330018816, 4444559976664662016, 84010278329827459072, 1289319649553742823424
Offset: 0

Views

Author

Ilya Gutkovskiy, May 23 2019

Keywords

Crossrefs

Cf. A008836, A053866, A205801, A306831.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Sum[x^(k^2) (2 + x^(k^2))/(2 k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[1/(1 - x^(2 k - 1))^(LiouvilleLambda[2 k - 1]/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} A053866(k)*x^k/k).
E.g.f.: Product_{k>=1} 1/(1 - x^(2*k-1))^(lambda(2*k-1)/(2*k-1)), where lambda() is the Liouville function (A008836).
Showing 1-4 of 4 results.

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

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