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

A289567 Coefficients in expansion of 1/E_6^(1/2).

Original entry on oeis.org

1, 252, 103572, 46355904, 21754545876, 10493652271032, 5153897870227008, 2563741466120209536, 1287429765611338091988, 651251466581383330576956, 331360676706818772917367912, 169399388595923901462013678656
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

1/E_k^(1/2): A289565 (k=2), A289566 (k=4), this sequence (k=6), A001943 (k=8), A289568 (k=10), A289569 (k=14).
E_6^(k/12): A289570 (k=-18), A000706 (k=-12), this sequence (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. A000706 (1/E_6), A288851, A289293 (E_6^(1/2)).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,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)^(-A288851(n)/2).
a(n) ~ c * exp(2*Pi*n) / sqrt(n), where c = 2^(5/2) * Gamma(3/4)^8 / (3*Pi^(5/2)) = 0.5480868931611627439175185425300450785609564636925943866686455998197... - Vaclav Kotesovec, Jul 09 2017, updated Mar 03 2018

A289247 Coefficients in expansion of 1/E_4^(1/8).

Original entry on oeis.org

1, -30, 3780, -616440, 111056910, -21135698280, 4165203862440, -840914061328320, 172810940671692900, -35998781800053352710, 7579904611028433074280, -1609957152292592382408360, 344417407415742189796786680, -74127324674775434904036905640
Offset: 0

Views

Author

Seiichi Manyama, Jul 08 2017

Keywords

Links

Crossrefs

E_4^(k/8): A001943 (k=-8), A289566 (k=-4), A295815 (k=-2), this sequence (k=-1), A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8).
Cf. A110163, A300147.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^(-1/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-A110163(n)/8).
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(7/8), where c = Pi^(3/2) / (2^(15/8) * 3^(1/4) * Gamma(1/3)^(9/4) * Gamma(9/8)) = 0.133402757019143151407904538533... - Vaclav Kotesovec, Jul 09 2017, updated Mar 05 2018
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 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

A378469 Coefficients in expansion of (1/E_4)^4.

Original entry on oeis.org

1, -960, 567360, -266138880, 108735481920, -40500351480960, 14114830665358080, -4678563821426250240, 1491145606587529742400, -460511820740945555286720, 138585483759128030100927360, -40812342463218781348220286720, 11800049457060387849887324117760, -3358272262154871467174772417214080
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 27 2024

Keywords

Comments

In general, for k > 0, the expansion of 1/(E_4)^k is asymptotic to (-1)^n * k * 2^(9*k) * Pi^(12*k) * n^(k-1) * exp(Pi*sqrt(3)*n) / (3^(2*k) * Gamma(1/3)^(18*k) * Gamma(k+1)).

Crossrefs

Cf. A001943 (k=1), A287933 (k=2), A378468 (k=3).
Cf. A289566 (k=1/2), A295815 (k=1/4), A289247 (k=1/8).
Cf. A004009, A289319.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1+240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])^(-4), {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * 34359738368 * Pi^48 * n^3 * exp(Pi*sqrt(3)*n) / (19683 * Gamma(1/3)^72).
Showing 1-6 of 6 results.

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