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 1-7 of 7 results.

A301562 Expansion of Product_{k>=0} (1 + x^(5*k+1))*(1 + x^(5*k+2)).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 6, 5, 4, 4, 6, 8, 8, 7, 7, 9, 12, 13, 11, 10, 12, 16, 19, 19, 17, 18, 23, 27, 27, 25, 25, 30, 37, 40, 38, 37, 42, 50, 55, 54, 52, 57, 68, 77, 78, 75, 78, 90, 102, 106, 104, 106, 120, 138, 146, 144, 145, 158
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 1 or 2 mod 5.

Examples

			a(13) = 3 because we have [12, 1], [11, 2] and [7, 6].

Links

Crossrefs

Cf. A035371, A047216, A107234, A107235, A203776, A219607, A280454, A281272, A301563, A301564, A301565, A301567, A301568, A301569, A301570.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[(1 + x^(5 k + 1)) (1 + x^(5 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x, x^5] QPochhammer[-x^2, x^5], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 2}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A047216(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(17/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301563 Expansion of Product_{k>=0} (1 + x^(5*k+1))*(1 + x^(5*k+3)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 4, 3, 4, 4, 4, 6, 4, 6, 7, 5, 9, 8, 8, 11, 9, 12, 12, 12, 16, 13, 17, 19, 17, 23, 21, 24, 27, 24, 32, 30, 32, 40, 35, 43, 45, 44, 53, 50, 59, 62, 61, 75, 70, 78, 87, 83, 99, 97, 105, 118, 112, 133, 134, 138, 159, 153
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 1 or 3 mod 5.

Examples

			a(14) = 3 because we have [13, 1], [11, 3] and [8, 6].

Links

Crossrefs

Cf. A035372, A047219, A107234, A107236, A203776, A219607, A280454, A281271, A301562, A301564, A301565, A301567, A301568, A301569, A301570.

Programs

  • Mathematica
    nmax = 72; CoefficientList[Series[Product[(1 + x^(5 k + 1)) (1 + x^(5 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 72; CoefficientList[Series[QPochhammer[-x, x^5] QPochhammer[-x^3, x^5], {x, 0, nmax}], x]
nmax = 72; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 3}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A047219(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(21/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

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

A301567 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 4, 1, 0, 2, 7, 7, 2, 0, 3, 10, 11, 4, 0, 4, 14, 17, 8, 1, 5, 19, 25, 13, 2, 6, 25, 36, 21, 4, 8, 33, 50, 33, 8, 10, 43, 69, 49, 14, 13, 55, 93, 71, 23, 17, 70, 124, 102, 37, 22, 88, 163, 142, 57, 30, 110, 212, 195, 85
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 1 mod 5.

Examples

			a(11) = 3 because we have [11], [10, 1] and [6, 5].

Links

Crossrefs

Cf. A008851, A035367, A036820, A203776, A219607, A280454, A301562, A301563, A301564, A301565, A301568, A301569, A301570.

Programs

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

Formula

G.f.: Product_{k>=2} (1 + x^A008851(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 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

A301570 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 2, 0, 0, 4, 7, 3, 0, 1, 7, 10, 4, 0, 2, 11, 14, 5, 0, 4, 17, 19, 6, 0, 8, 25, 25, 8, 1, 13, 36, 33, 10, 2, 21, 50, 43, 12, 4, 33, 69, 55, 15, 8, 49, 93, 70, 18, 14, 71, 124, 88, 23, 23, 102, 163, 110, 29, 37
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 4 mod 5.

Examples

			a(14) = 3 because we have [14], [10, 4] and [9, 5].

Links

Crossrefs

Cf. A035370, A036820, A047208, A203776, A219607, A281243, A301562, A301563, A301564, A301565, A301567, A301568, A301569.

Programs

  • Mathematica
    nmax = 76; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 76; CoefficientList[Series[x QPochhammer[-1, x^5] QPochhammer[-x^(-1), x^5]/(2 (1 + x)), {x, 0, nmax}], x]
nmax = 76; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} (1 + x^A047208(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(41/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018
Showing 1-7 of 7 results.

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