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 1-8 of 8 results.

A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 2, -1, 0, -1, 2, -1, 0, -1, 3, -2, 0, -1, 3, -3, 1, -1, 4, -4, 1, -1, 4, -5, 2, -1, 5, -7, 3, -1, 5, -8, 5, -2, 6, -10, 6, -2, 6, -12, 9, -3, 7, -14, 11, -4, 7, -16, 15, -6, 8, -19, 18, -8, 9, -21, 23, -11, 10, -24
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), this sequence (m=4), A284314 (m=5).
Cf. A050449, A169975, A284316.

Programs

  • Maple
    V:= Vector(100):
V[1]:= 1:
for k from 0 to 24 do
  V[4*k+2..100]:= V[4*k+2..100] - V[1..99-4*k]
od:
convert(V,list); # Robert Israel, May 03 2017
  • Mathematica
    CoefficientList[Series[Product[1 - x^(4k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(4*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A050449(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. A284316. - Peter Bala, Nov 28 2020

A284312 Expansion of Product_{k>=0} (1 - x^(3*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 0, -1, 1, 0, -1, 2, -1, -1, 2, -1, -1, 3, -2, -1, 3, -3, 0, 4, -4, 0, 4, -5, 1, 5, -7, 2, 5, -8, 4, 5, -10, 5, 5, -12, 8, 5, -14, 10, 4, -16, 14, 3, -19, 17, 1, -20, 22, -1, -23, 26, -4, -25, 33, -8, -27, 38, -13, -28, 46, -19, -30, 53, -26, -29
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), this sequence (m=3), A284313 (m=4), A284314 (m=5).
Cf. A078181, A261612.

Programs

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

Formula

a(n) = -(1/n)*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.

A359936 Expansion of Product_{k>=0} (1 - x^(k^2+1)) in powers of x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jan 19 2023

Keywords

Links

Crossrefs

Cf. A002522, A081362, A276516, A284312, A284313, A284314, A284585, A284499, A357911, A359937.

Programs

  • PARI
    my(N=100, x='x+O('x^N)); Vec(prod(k=0, sqrtint(N), 1-x^(k^2+1)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, sumdiv(j, d, issquare(d-1)*d)*v[i-j+1])/i); v;

Formula

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A359937(k) * a(n-k).

A284320 Expansion of Product_{k>=0} (1 - x^(5*k+3)) in powers of x.

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 2, 0, -1, -1, 0, 2, 0, -1, -1, 0, 3, 0, -1, -2, 0, 3, 0, -1, -3, 0, 4, 1, -1, -4, 0, 4, 1, -1, -5, 0, 5, 2, -1, -7, 0, 5, 3, -1, -8, 0, 6, 5, -1, -10, -1, 6, 6, -1, -12, -1, 7, 9, -1, -14, -2, 7, 11
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(5*k+m)): A284314 (m=1), A284319 (m=2), this sequence (m=3), A284317 (m=4).
Cf. A281271, A284281.

Programs

  • Mathematica
    CoefficientList[Series[Product[1 - x^(5k + 3), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *) (* or *)
a[0]=1; a[n_]:=a[n]= -(1/n) Sum[ a[n-k] DivisorSum[k, # &, Mod[#,5] == 3 &], {k, n}]; a /@ Range[0, 100] (* Giovanni Resta, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(5*k + 3)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284281(k)*a(n-k), a(0) = 1.

A284319 Expansion of Product_{k>=0} (1 - x^(5*k+2)) in powers of x.

Original entry on oeis.org

1, 0, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, 2, 0, -1, -1, 0, 2, 0, -1, -1, 0, 3, 0, -2, -1, 0, 3, 0, -3, -1, 1, 4, 0, -4, -1, 1, 4, 0, -5, -1, 2, 5, 0, -7, -1, 3, 5, 0, -8, -1, 5, 6, -1, -10, -1, 6, 6, -1, -12, -1, 9, 7, -2, -14, -1, 11, 7, -3
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Crossrefs

Cf. Product_{k>=0} (1 - x^(5*k+m)): A284314 (m=1), this sequence (m=2), A284320 (m=3), A284317 (m=4).
Cf. A281272, A284280.

Programs

  • Mathematica
    CoefficientList[Series[Product[1 - x^(5k + 2), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(5*k + 2)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284280(k)*a(n-k), a(0) = 1.

A284321 Expansion of Product_{k>=0} (1 - x^(5*k+1))*(1 - x^(5*k+4)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 1, 0, -1, 2, -2, 1, 1, -2, 3, -3, 2, 0, -3, 5, -5, 3, 1, -5, 7, -7, 4, 1, -7, 11, -11, 6, 2, -10, 15, -15, 9, 2, -14, 22, -22, 12, 4, -20, 30, -29, 17, 4, -27, 42, -41, 23, 7, -37, 55, -54, 31, 8, -49, 76, -74, 41, 12, -66, 99, -96, 55, 14
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. A003114, A284150, A284322, A284314, A284317.
Cf. Product_{k>=0} (1 - x^(m*k+1))*(1 - x^(m*k+m-1)): A137569 (m=3), A081362 (m=4), this sequence (m=5), A109389 (m=6).

Programs

  • Mathematica
    CoefficientList[Series[Product[(1 - x^(5k + 1)) ( 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 + 1)) * (1 - x^(5*k + 4))) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1.

A357911 Expansion of Product_{k>=0} (1 - x^(11*k+1)) in powers of x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jan 17 2023

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), A284313 (m=4), A284314 (m=5), A284585 (m=6), A284499 (m=7), this sequence (m=11).
Cf. A357912.

Programs

  • PARI
    my(N=100, x='x+O('x^N)); Vec(prod(k=0, N, 1-x^(11*k+1)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, sumdiv(j, d, (Mod(d, 11)==1)*d)*v[i-j+1])/i); v;

Formula

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A357912(k) * a(n-k).

A374079 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-4)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, -1, 2, -2, 2, -2, 1, 0, -1, 1, -1, 0, 2, -3, 4, -4, 3, -1, -1, 2, -3, 2, 1, -4, 6, -7, 7, -4, 0, 3, -5, 5, -2, -3, 8, -11, 12, -9, 3, 3, -8, 10, -7, 0, 8, -15, 19, -17, 9, 1, -10, 16, -15, 6, 7, -19, 28, -29, 20, -5, -11, 23, -26, 17, 1, -21
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109697, A280454, A284314, A374064, A374065, A374076, A374077, A374078.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]
Showing 1-8 of 8 results.

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