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.

Previous Showing 11-14 of 14 results.

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

A294181 Coefficients in expansion of E_2/E_4.

Original entry on oeis.org

1, -264, 61128, -14107296, 3255470952, -751247454384, 173361309784992, -40005651284526912, 9231887649122522280, -2130392752758423726312, 491619206548389935051568, -113448303808924351510423008, 26179851123971817380111236128
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2018

Keywords

Links

Crossrefs

Cf. A001943, A004009 (E_4), A006352 (E_2), A288877.
E_k/E_{k+2}: this sequence (k=2), A294182 (k=4), A294183 (k=6).

Programs

  • Mathematica
    terms = 13;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E2[x]/E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

Formula

Convolution inverse of A288877.
a(n) ~ (-1)^n * 1024 * Pi^11 * exp(Pi*sqrt(3)*n) / (3^(3/2) * Gamma(1/3)^18). - Vaclav Kotesovec, Jun 03 2018

A378468 Coefficients in expansion of (1/E_4)^3.

Original entry on oeis.org

1, -720, 339120, -132039360, 46081214640, -14974899930720, 4627836408778560, -1377759164154871680, 398508058352289409200, -112648427646194257313040, 31252327416307233967209120, -8536592939398421710286859840, 2301363255613811638678456000320, -613491781086725734777586106900960
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 27 2024

Keywords

Crossrefs

Cf. A004009, A001943, A287933, A378469.

Programs

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

Formula

a(n) ~ (-1)^n * 67108864 * Pi^36 * n^2 * exp(Pi*sqrt(3)*n) / (729 * Gamma(1/3)^54).

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).
Previous Showing 11-14 of 14 results.

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