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 1-4 of 4 results.

A301568 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-3)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 2, 2, 0, 5, 0, 4, 2, 1, 7, 0, 7, 3, 2, 10, 0, 11, 4, 4, 14, 0, 17, 5, 8, 19, 1, 25, 6, 13, 25, 2, 36, 8, 21, 33, 4, 50, 10, 33, 43, 8, 69, 12, 49, 55, 14, 93, 16, 71, 70, 23, 124, 20, 102, 88, 37, 163, 26, 142, 110, 57, 212
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 2 mod 5.

Examples

			a(12) = 3 because we have [12], [10, 2] and [7, 5].

Links

Crossrefs

Cf. A035368, A036822, A047215, A203776, A219607, A281272, A301562, A301563, A301564, A301565, A301567, A301569, A301570.

Programs

  • Mathematica
    nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[x^3 QPochhammer[-1, x^5] QPochhammer[-x^(-3), x^5]/(2 (1 + x) (1 - x + x^2)), {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 2}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A047215(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(33/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301569 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 3, 0, 2, 2, 0, 5, 0, 2, 4, 0, 7, 1, 3, 7, 0, 10, 2, 4, 11, 0, 14, 4, 5, 17, 0, 19, 8, 6, 25, 1, 25, 13, 8, 36, 2, 33, 21, 10, 50, 4, 43, 33, 12, 69, 8, 55, 49, 15, 93, 14, 70, 71, 19, 124, 23, 88, 102, 24, 163, 37, 110, 142, 31
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 3 mod 5.

Examples

			a(13) = 3 because we have [13], [10, 3] and [8, 5].

Links

Crossrefs

Cf. A035369, A036822, A047218, A203776, A219607, A281271, A301562, A301563, A301564, A301565, A301567, A301568, A301570.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[x^2 QPochhammer[-1, x^5] QPochhammer[-x^(-2), x^5]/(2 (1 + x^2)), {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 3}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} (1 + x^A047218(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(37/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A346797 Number of partitions of n into parts congruent to 0, 2 or 5 (mod 7).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 7, 4, 9, 6, 10, 11, 11, 17, 13, 22, 19, 25, 29, 28, 42, 34, 53, 46, 61, 67, 69, 92, 83, 115, 109, 133, 149, 152, 198, 182, 243, 233, 282, 309, 324, 398, 385, 485, 483, 563, 621, 648, 784, 768, 944, 947, 1096, 1194, 1262
Offset: 0

Views

Author

Ludovic Schwob, Aug 04 2021

Keywords

Examples

			For n=17 the a(17)=6 solutions are 2+2+2+2+2+2+5, 2+2+2+2+2+7, 2+2+2+2+9, 2+5+5+5, 5+5+7 and 5+12.

Links

Crossrefs

Cf. A000041, A047386, A274830, A006950, A036820, A036822, A195848, A035363, A195849, A346798.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-2))(1 - x^(7*k-5))),{k,52}],{x,0,52}],x] (* Stefano Spezia, Aug 04 2021 *)

Formula

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-2))*(1 - x^(7*k-5))).
a(n) = a(n-2) + a(n-5) - a(n-11) - a(n-17) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 2, 5, 11, 17, ... is the sequence A274830.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(3*Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A346798 Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 3, 6, 4, 4, 8, 9, 6, 10, 15, 12, 12, 21, 22, 18, 25, 36, 30, 32, 48, 52, 45, 60, 78, 72, 75, 105, 113, 105, 130, 166, 156, 166, 218, 236, 224, 274, 332, 325, 345, 436, 469, 462, 544, 649, 644, 688, 839, 907, 903, 1051
Offset: 0

Views

Author

Ludovic Schwob, Aug 04 2021

Keywords

Examples

			For n=19 the a(19)=6 solutions are 3+3+3+3+3+4, 3+3+3+3+7, 3+3+3+10, 3+4+4+4+4, 4+4+4+7, and 4+4+11.

Links

Crossrefs

Cf. A000041, A047342, A057570, A006950, A036820, A036822, A195848, A035363, A195849, A346797.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-3))(1 - x^(7*k-4))),{k,55}],{x,0,55}],x] (* Stefano Spezia, Aug 04 2021 *)

Formula

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-3))*(1 - x^(7*k-4))).
a(n) = a(n-3) + a(n-4) - a(n-13) - a(n-15) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 4, 13, 15, ... is the sequence A057570.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021
Showing 1-4 of 4 results.

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