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 11-17 of 17 results.

A284317 Expansion of Product_{k>=0} (1 - x^(5*k+4)) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 3, -1, 0, 0, -2, 3, -1, 0, 0, -3, 4, -1, 0, 1, -4, 4, -1, 0, 1, -5, 5, -1, 0, 2, -7, 5, -1, 0, 3, -8, 6, -1, 0, 5, -10, 6, -1, -1, 6, -12, 7, -1, -1, 9, -14
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), A284315 (m=3), A284316 (m=4), this sequence (m=5).
Cf. A281243, A284103.

Programs

  • Maple
    S:= series(mul(1-x^(5*k+4),k=0..200),x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 27 2017
  • Mathematica
    CoefficientList[Series[Product[1 - x^(5k + 4), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(5*k + 4)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1.
G.f. is the QPochhammer symbol (x^4;x^5)infinity. - _Robert Israel, Mar 27 2017

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

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

A339087 Number of compositions (ordered partitions) of n into distinct parts congruent to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 4, 1, 0, 0, 6, 4, 1, 0, 0, 6, 6, 1, 0, 0, 12, 6, 1, 0, 0, 18, 8, 1, 0, 24, 24, 8, 1, 0, 24, 30, 10, 1, 0, 48, 42, 10, 1, 0, 72, 48, 12, 1, 0, 120, 60, 12, 1, 120, 144, 72, 14, 1, 120, 216, 84, 14, 1, 240
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2020

Keywords

Examples

			a(27) = 6 because we have [14, 9, 4], [14, 4, 9], [9, 14, 4], [9, 4, 14], [4, 14, 9] and [4, 9, 14].

Links

Crossrefs

Cf. A016897, A017827, A032020, A032021, A109700, A281243, A337547, A337548, A339059, A339060, A339086, A339088, A339089.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[k! x^(k (5 k + 3)/2)/Product[1 - x^(5 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)).

A374076 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 2, -1, -1, 1, -2, 2, 0, -1, 2, -3, 2, 0, -2, 3, -3, 2, 1, -3, 4, -4, 2, 2, -4, 5, -5, 1, 3, -6, 7, -5, 1, 5, -8, 8, -6, -1, 8, -10, 11, -6, -3, 10, -14, 12, -5, -6, 15, -17, 14, -4, -10, 19, -21, 15, -1, -15, 25, -25
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109700, A281243, A284317, A374064, A374065, A374077, A374078, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 4 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A284093 Expansion of Product_{k>=1} (1 + x^(8*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 1, 5, 5, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 20 2017

Keywords

Comments

Number of partitions into distinct parts 8*k-1.

Links

Crossrefs

Cf. Product_{k>=1} (1 + x^(m*k-1)): A262928 (m=3), A147599 (m=4), A281243 (m=5), A281244 (m=6), A281245 (m=7), this sequence (m=8).

Programs

  • Mathematica
    CoefficientList[Series[Product[(1 + x^(8*k - 1)) , {k, 1, 91}], {x, 0, 91}], x] (* Indranil Ghosh, Mar 20 2017 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 8] == 7, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 20 2017 *)
  • PARI
    Vec(prod(k=1, 91, (1 + x^(8*k - 1))) + O(x^92)) \\ Indranil Ghosh, Mar 20 2017

Formula

a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(21/8) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(384*sqrt(6)) - 3*sqrt(3/2)/(2*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017
G.f.: Sum_{k>=0} x^(k*(4*k + 3)) / Product_{j=1..k} (1 - x^(8*j)). - Ilya Gutkovskiy, Nov 24 2020
Previous Showing 11-17 of 17 results.

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