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.

A284314 Expansion of Product_{k>=0} (1 - x^(5*k+1)) in powers of x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), A284313 (m=4), this sequence (m=5).
Cf. A280454, A284097.

Programs

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

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284097(k)*a(n-k), a(0) = 1.

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).

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

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).

A111330 Let qf(a,q) = Product_{j >= 0} (1-a*q^j); g.f. is qf(q,q^4)/qf(q^3,q^4).

Original entry on oeis.org

1, -1, 0, 1, -1, -1, 2, 0, -2, 1, 1, -1, -1, 1, 2, -2, -2, 3, 1, -4, 0, 5, -1, -5, 2, 5, -4, -5, 6, 4, -6, -4, 7, 4, -10, -2, 12, 0, -13, 2, 13, -4, -14, 6, 17, -10, -17, 14, 15, -17, -15, 21, 15, -26, -13, 31, 9, -35, -5, 39, 2, -44, 3, 49, -12, -52, 21, 53, -27, -55, 35, 57, -47, -57, 59, 55, -69, -52, 80, 49, -95, -43, 110, 34, -122
Offset: 0

Views

Author

N. J. A. Sloane, Nov 09 2005

Keywords

Crossrefs

Cf. A111317, A111335, A111374, A224704, A284313, A284316.

Formula

From Peter Bala, Nov 28 2020: (Start)
O.g.f.: A(x) = F(x)/G(x) where F(x) = Product_{k >= 0} 1 - x^(4*k+1) (see A284313) and G(x) = Product_{k >= 0} 1 - x^(4*k+3) (see A284316).
Continued fraction representations: A(x) = 1 - x/(1 + x^2 - x^3/(1 + x^4 - x^5/(1 + x^6 - ... ))).
A(x) = 1 - x/(1 - x^2*(x - 1)/(1 - x^5/(1 - x^4*(x^3 - 1)/(1 - x^9/(1 - x^6*(x^5 - 1)/(1 - ... )))))). Cf. A224704. (End)

A374081 Expansion of Product_{k>=1} (1 - x^(4*k-3)) * (1 - x^(4*k)).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Links

Crossrefs

Cf. A010815, A035362, A042948, A081362, A082303, A106459, A137569, A274719, A284313, A301504, A374058, A374060, A374080.

Programs

  • Mathematica
    nmax = 85; CoefficientList[Series[Product[(1 - x^(4 k - 3)) (1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[DivisorSum[k, # &, Or[Mod[#, 4] == 0, Mod[#, 4] == 1] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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