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 1-5 of 5 results.

A282919 a(n) = A000041(49*n + 47).

Original entry on oeis.org

124754, 118114304, 24908858009, 2366022741845, 133978259344888, 5234371069753672, 154043597379576030, 3617712763867604423, 70593393646562135510, 1178875491155735802646, 17229817230617210720599, 224282898599046831034631, 2636785814481962651219075
Offset: 0

Views

Author

Seiichi Manyama, Feb 24 2017

Keywords

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 179.

Links

Crossrefs

Cf. A160528, A282920, A282921, A282922, A282923, A282924, A282925, A282926, A282927, A282928, A282929, A282930, A282931, A282932.
Cf. A000041, A213261 (p(7*n + 5)), A277958, A278559 (p(25*n + 24)), this sequence (p(49*n + 47)).

Programs

  • Mathematica
    Table[PartitionsP[49n+47],{n,0,12}] (* Indranil Ghosh, Feb 25 2017 *)
  • PARI
    a(n) = numbpart(49*n+47); \\ Indranil Ghosh, Feb 25 2017

Formula

a(n) = A213261(7*n + 6) = A000041(49*n + 47).
a(n) = 2546 * 7^2 * A160528(n) + 48934 * 7^4 * A282920(n-1) + 1418989 * 7^5 * A282921(n-2) + 2488800 * 7^7 * A282922(n-3) + 2394438 * 7^9 * A282923(n-4) + 1437047 * 7^11 * A282924(n-5) + 4043313 * 7^12 * A282925(n-6) + 161744 * 7^15 * A282926(n-7) + 32136 * 7^17 * A282927(n-8) + 31734 * 7^18 * A282928(n-9) + 3120 * 7^20 * A282929(n-10) + 204 * 7^22 * A282930(n-11) + 8 * 7^24 * A282931(n-12) + 7^25 * A282932(n-13) for n >= 13.

A278555 Expansion of Product_{n>=1} (1 - x^(5*n))^12/(1 - x^n)^13 in powers of x.

Original entry on oeis.org

1, 13, 104, 637, 3276, 14808, 60541, 228124, 803010, 2667054, 8422715, 25446304, 73907808, 207209614, 562673618, 1484147681, 3811882087, 9553588317, 23407932874, 56161135485, 132132608899, 305240006266, 693150485885, 1548871015291, 3408852663762, 7395582677152
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Comments

In general, if m>0 and g.f. = Product_{k>=1} (1 - x^(5*k))^m/(1 - x^k)^(m+1) then a(n) ~ sqrt(4*m+5) * exp(Pi*sqrt(2*(4*m+5)*n/15)) / (4*sqrt(3)*5^((m+1)/2)*n). - Vaclav Kotesovec, Nov 24 2016

Links

Crossrefs

Cf. A160460, A278556, A278557, A278558, A278559.

Programs

  • Mathematica
    CoefficientList[ Series[ Product[(1 - x^(5n))^12/(1 - x^n)^13, {n, 25}],
{x, 0, 25}], x] (* Robert G. Wilson v, Nov 23 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^12/(1 - x^n)^13.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*a(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(53/15)*exp(sqrt(106*n/15)*Pi)/(62500*n). - Vaclav Kotesovec, Nov 24 2016

A278556 Expansion of Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19 in powers of x.

Original entry on oeis.org

1, 19, 209, 1710, 11495, 66862, 347339, 1645875, 7221520, 29668595, 115116233, 424720338, 1498263563, 5076482415, 16583497160, 52399330389, 160586833362, 478482249548, 1388989067820, 3935549005725, 10901608510397, 29565343541110, 78604103339462
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), this sequence (k=18), A278557 (k=24), A278558 (k=30).

Programs

  • Mathematica
    CoefficientList[ Series[ Product[(1 - x^(5n))^18/(1 - x^n)^19, {n, 22}], {x, 0, 22}], x] (* Robert G. Wilson v, Nov 24 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*a(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(77/15) * exp(Pi*sqrt(154*n/15)) / (7812500*n). - Vaclav Kotesovec, Nov 28 2016

A278557 Expansion of Product_{n>=1} (1 - x^(5*n))^24/(1 - x^n)^25 in powers of x.

Original entry on oeis.org

1, 25, 350, 3575, 29575, 209381, 1312075, 7443825, 38854075, 188836375, 862496902, 3729343275, 15356254650, 60511763600, 229125615600, 836555203223, 2953900713000, 10113407774450, 33649438734125, 109017926343725, 344525085375315, 1063718962906450
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Links

Crossrefs

Cf. A160460, A278555, A278556, A278558, A278559.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^24/(1 - x^k)^25, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^24/(1 - x^n)^25.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*A278556(n-2) + 5^10*6*a(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(101/15) * exp(Pi*sqrt(202*n/15)) / (976562500*n). - Vaclav Kotesovec, Nov 28 2016

A278558 Expansion of Product_{n>=1} (1 - x^(5*n))^30/(1 - x^n)^31 in powers of x.

Original entry on oeis.org

1, 31, 527, 6448, 63240, 526443, 3852742, 25380847, 153068700, 855816380, 4479330091, 22117432019, 103672066076, 463698703204, 1987628351600, 8195086588810, 32603090921532, 125497791966435, 468512597653134, 1699911932127300, 6005651320362628, 20693956328627358
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2016

Keywords

Comments

In general, if m>0 and g.f. = Product_{k>=1} (1 - x^(5*k))^m/(1 - x^k)^(m+1) then a(n) ~ sqrt(4*m+5) * exp(Pi*sqrt(2*(4*m+5)*n/15)) / (4*sqrt(3)*5^((m+1)/2)*n). - Vaclav Kotesovec, Nov 28 2016

Links

Crossrefs

Cf. A160460, A278555, A278556, A278557, A278559.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^30/(1 - x^k)^31, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^30/(1 - x^n)^31.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*a(n-4) for n >= 4.
a(n) ~ exp(Pi*5*sqrt(2*n/3)) / (24414062500*sqrt(3)*n). - Vaclav Kotesovec, Nov 28 2016
Showing 1-5 of 5 results.

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