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 1-6 of 6 results.

A289565 Coefficients in expansion of 1/E_2^(1/2).

Original entry on oeis.org

1, 12, 252, 5664, 133356, 3224952, 79387488, 1978996416, 49797787788, 1262193008556, 32177428972632, 824182154521056, 21193138994244960, 546767126418119352, 14146104826919725632, 366887630982365262144, 9535791498480146879436
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): this sequence (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
Cf. A288816 (1/E_2), A288968, A289291 (E_2^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A288968(n)/2).
a(n) ~ c / (sqrt(n) * r^n), where r = A211342 = 0.03727681029645165815098078565... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24 and c = 0.535261044779387956394739769118415667289349331646288208543596374426... - Vaclav Kotesovec, Jul 09 2017

A289566 Coefficients in expansion of 1/E_4^(1/2).

Original entry on oeis.org

1, -120, 20520, -3934560, 793510440, -164694615120, 34824089129760, -7460017581785280, 1613575314347164200, -351613291994820018840, 77073167391611232305520, -16975579813113940564868640, 3753822590560913900129106720
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), this sequence (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
Cf. A001943 (1/E_4), A110163, A289292 (E_4^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A110163(n)/2).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / sqrt(n), where c = 3^(7/2) * Gamma(2/3)^9 / (2^(9/2) * Pi^(7/2)) = 0.5756695813762774104492155417156662666189119445257965... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

A289568 Coefficients in expansion of 1/E_10^(1/2).

Original entry on oeis.org

1, 132, 93852, 35163744, 18119136156, 8462089683432, 4234179302847648, 2096050696254014016, 1057219212439789539228, 534730176137991079392036, 272470142855167873443179352, 139363825115618499934478625696
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), this sequence (k=10), A289569 (k=14).
Cf. A285836 (1/E_10), A289024, A289294 (E_10^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 264*Sum[DivisorSigma[9, k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A289024(n)/2).
a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 0.4542595790370690606664796229968194763901027924111318430568304678613... = 2^(7/2) * Gamma(3/4)^12 / (3^(3/2) * Pi^(7/2)). - Vaclav Kotesovec, Jul 09 2017, updated Mar 07 2018

A289569 Coefficients in expansion of 1/E_14^(1/2).

Original entry on oeis.org

1, 12, 98532, 22675584, 16099478436, 6580135809432, 3539736295913088, 1699883073000957696, 871767496424764386468, 438331617201642108107916, 224266585355757815798085192, 114622723650418140746841457536, 58945651172799536532104421386880
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), A289567 (k=6), A001943 (k=8), A289568 (k=10), this sequence (k=14).
Cf. A287964 (1/E_14), A289029, A289295 (E_14^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 24*Sum[DivisorSigma[13, k]*x^k, {k, 1, nmax}])^(-1/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A289029(n)/2).
a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 0.3764946174077880880364705796802173599460310621830541667074693852949... = 2^(9/2) * Gamma(3/4)^16 / (9 * Pi^(9/2)). - Vaclav Kotesovec, Jul 09 2017, updated Mar 07 2018

A289570 Coefficients in expansion of 1/E_6^(3/2).

Original entry on oeis.org

1, 756, 501228, 311671584, 187266950892, 110121960638088, 63808586297102304, 36578013578688141504, 20797655630223547290348, 11749541312124028845092052, 6603568491137827506152966712, 3695593478842608407829235523808
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

E_6^(k/12): this sequence (k=-18), A000706 (k=-12), A289567 (k=-6), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-3/2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-3*A288851(n)/2).
a(n) ~ c * exp(2*Pi*n) * sqrt(n), where c = 2^(17/2) * Gamma(3/4)^24 / (27 * Pi^(13/2)) = 1.0344943380746471723299237298670710161068814236907171661035... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018

A289540 Coefficients in expansion of 1/E_6^(1/12).

Original entry on oeis.org

1, 42, 12852, 4780104, 1974512526, 863778376440, 391960077239304, 182430901827757632, 86505196617272556900, 41607881477457256661154, 20239469012268054187498440, 9935363620927698868439915544, 4914082482014906612773260362232
Offset: 0

Views

Author

Seiichi Manyama, Jul 15 2017

Keywords

Links

Crossrefs

E_6^(k/12): A289570 (k=-18), A000706 (k=-12), A289567 (k=-6), this sequence (k=-1), A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973, A288851, A299503.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(-1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 26 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(5/12) * Gamma(3/4)^(4/3) / (3^(1/6) * Pi^(1/3) * Gamma(1/12)) = 0.08654217651555778130817946575840803466... - Vaclav Kotesovec, Jul 26 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
Showing 1-6 of 6 results.

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