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.

A301564 Expansion of Product_{k>=0} (1 + x^(5*k+2))*(1 + x^(5*k+4)).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 2, 1, 2, 2, 1, 3, 1, 3, 3, 2, 5, 2, 6, 3, 5, 6, 4, 8, 5, 8, 8, 7, 12, 7, 13, 11, 11, 16, 11, 19, 14, 19, 21, 17, 27, 20, 27, 28, 26, 36, 28, 40, 37, 38, 49, 39, 55, 49, 55, 64, 55, 76, 65, 78, 84, 78, 100, 87, 107, 109, 107, 134, 116, 145
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 2 or 4 mod 5.

Examples

			a(16) = 3 because we have [14, 2], [12, 4] and [9, 7].

Links

Crossrefs

Cf. A035373, A047211, A107235, A107237, A203776, A219607, A281243, A281272, A301562, A301563, A301565, A301567, A301568, A301569, A301570.

Programs

  • Mathematica
    nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 2)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[QPochhammer[-x^2, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{2, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A047211(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301565 Expansion of Product_{k>=0} (1 + x^(5*k+3))*(1 + x^(5*k+4)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 4, 4, 3, 3, 4, 6, 6, 4, 4, 7, 9, 7, 6, 8, 11, 12, 10, 9, 12, 16, 16, 14, 14, 19, 23, 22, 19, 21, 27, 31, 29, 26, 31, 40, 42, 38, 38, 45, 53, 55, 51, 52, 63, 73, 73, 69, 73, 87, 97, 95, 91, 100, 118, 128
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 3 or 4 mod 5.

Examples

			a(17) = 3 because we have [14, 3], [13, 4] and [9, 8].

Links

Crossrefs

Cf. A035374, A047204, A107236, A107237, A203776, A219607, A281243, A281271, A301562, A301563, A301564, A301567, A301568, A301569, A301570.

Programs

  • Mathematica
    nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 3)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[QPochhammer[-x^3, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{3, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A047204(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(33/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A107234 Expansion of 1 / Product_{n>=0} (1-q^(5n+1))(1-q^(5n+2))(1-q^(5n+3)).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 10, 14, 18, 23, 29, 38, 47, 60, 74, 92, 112, 139, 168, 205, 247, 298, 356, 429, 509, 607, 718, 850, 1000, 1180, 1381, 1620, 1890, 2206, 2564, 2983, 3453, 4000, 4618, 5330, 6133, 7059, 8097, 9289, 10630, 12159, 13877
Offset: 0

Views

Author

Ralf Stephan, May 13 2005

Keywords

Comments

a(n) is the number of partitions of n into parts 5k+1, 5k+2 or 5k+3. - George Beck, Aug 09 2020

Links

Crossrefs

Cf. A035959, A107235, A107236, A107237.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+1))*(1 - x^(5*k+2))*(1 - x^(5*k+3)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

Formula

a(n) ~ Pi^(1/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(4/5) * 2^(3/5) * 5^(9/10) * n^(3/5)). - Vaclav Kotesovec, Jan 07 2021

A107235 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))*(1 - q^(5n+2))*(1 - q^(5n+4)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 19, 23, 31, 37, 48, 57, 73, 86, 109, 128, 159, 187, 229, 269, 326, 382, 458, 535, 638, 742, 879, 1019, 1200, 1388, 1625, 1875, 2185, 2514, 2916, 3347, 3868, 4427, 5099, 5822, 6683, 7614, 8712, 9904, 11301, 12821, 14589
Offset: 0

Views

Author

Ralf Stephan, May 13 2005

Keywords

Links

Crossrefs

Cf. A035959, A107234, A107236, A107237.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+1))*(1 - x^(5*k+2))*(1 - x^(5*k+4)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

Formula

a(n) ~ Pi^(2/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(3/5) * 2^(7/10) * 5^(4/5) * n^(7/10)). - Vaclav Kotesovec, Jan 07 2021

A107236 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))*(1 - q^(5n+3))*(1 - q^(5n+4)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 8, 11, 13, 16, 22, 26, 32, 40, 49, 59, 73, 87, 105, 126, 151, 178, 214, 252, 297, 351, 413, 481, 566, 658, 767, 892, 1034, 1195, 1386, 1595, 1838, 2114, 2429, 2781, 3189, 3642, 4160, 4744, 5404, 6141, 6986, 7921, 8980
Offset: 0

Views

Author

Ralf Stephan, May 13 2005

Keywords

Links

Crossrefs

Cf. A035959, A107234, A107235, A107237.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+1))*(1 - x^(5*k+3))*(1 - x^(5*k+4)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

Formula

a(n) ~ Pi^(3/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(2/5) * 2^(4/5) * 5^(7/10) * n^(4/5)). - Vaclav Kotesovec, Jan 07 2021
Showing 1-5 of 5 results.

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