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.

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

A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 2, -1, 0, -1, 2, -1, 0, -1, 3, -1, 0, -2, 3, -1, 0, -3, 4, -1, 1, -4, 4, -1, 1, -5, 5, -1, 2, -7, 5, -1, 3, -8, 6, -1, 5, -10, 6, -2, 6, -12, 7, -2, 9, -14, 7, -3, 11, -16, 8, -4, 15, -19, 8, -6, 18, -21, 9
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), this sequence (m=4), A284317 (m=5).
Cf. A050452, A147599, A284313, A111330.

Programs

  • Mathematica
    CoefficientList[Series[Product[1 - x^(4k + 3), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(4*k+3)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1.
O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n+1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284313. - Peter Bala, Nov 28 2020

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

Original entry on oeis.org

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A374060 Expansion of Product_{k>=1} (1 - x^(3*k-1)) * (1 - x^(3*k)).

Original entry on oeis.org

1, 0, -1, -1, 0, 0, -1, 1, 1, 0, 0, 1, 0, 0, 1, 0, -1, 0, 0, -1, 0, 1, -1, -1, 0, -1, 0, 1, 0, -1, 1, 0, -1, 1, 1, -1, 0, 1, 0, 1, 1, -1, 0, 1, -1, -1, 2, 0, -1, 1, 0, -1, 1, 0, -2, 0, 0, -1, 1, 1, -2, 0, 1, -2, 0, 2, -1, -1, 1, -1, -1, 2, -1, -1, 2, 0, -1, 2, 1, -2, 1, 0, -2, 2, 1, -2
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Links

Crossrefs

Cf. A010815, A035361, A035382, A081362, A082050, A132462, A137569, A274719, A284315, A374058.

Programs

  • Mathematica
    nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 1)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

Formula

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082050(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} A035361(k) * a(n-k).
a(n) = Sum_{k=0..n} A010815(k) * A035382(n-k).
Showing 1-4 of 4 results.

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