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.

A291983 Expansion of 1/((1+x)*(1+x^2)*(1+x^3)).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Links

Crossrefs

Cf. A112689, A291954, A291984, A291985.

Programs

  • Mathematica
    CoefficientList[Series[1/((1+x)(1+x^2)(1+x^3)),{x,0,70}],x] (* or *) LinearRecurrence[ {-1,-1,-2,-1,-1,-1},{1,-1,0,-1,2,-1},80] (* Harvey P. Dale, Jun 12 2022 *)
  • PARI
    Vec(1/((1+x)*(1+x^2)*(1+x^3)) + O(x^100))

A291984 Expansion of 1/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)).

Original entry on oeis.org

1, -1, 0, -1, 1, 0, 1, -1, 1, -2, 1, -1, 2, -1, 1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 3, -3, 2, -3, 3, -2, 3, -3, 3, -4, 3, -3, 4, -3, 3, -4, 4, -4, 4, -4, 4, -4, 4, -4, 5, -5, 4, -5, 5, -4, 5, -5, 5, -6, 5, -5, 6, -5, 5, -6, 6, -6, 6, -6, 6, -6, 6, -6, 7, -7, 6, -7
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Links

Crossrefs

Cf. A291954, A291983, A291985.

Programs

  • PARI
    Vec(1/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) + O(x^100))

A291985 Expansion of 1/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)).

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 2, -1, 2, -3, 2, -3, 3, -3, 4, -4, 5, -5, 5, -6, 6, -7, 7, -7, 9, -9, 9, -10, 10, -11, 12, -12, 13, -14, 14, -15, 16, -16, 17, -18, 19, -20, 20, -21, 22, -23, 24, -24, 26, -27, 27, -29, 29, -30, 32, -32, 34, -35, 35, -37, 38, -39, 40, -41, 43
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2017

Keywords

Links

Crossrefs

Cf. A291954, A291983, A291984.

Programs

  • PARI
    Vec(1/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)) + O(x^100))

A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Apr 07 2025

Keywords

Examples

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  1;
  0, 1, 3,  2;
  0, 1, 4,  3;
  0, 1, 4,  4, 1;
  0, 1, 5,  5, 1;
  0, 1, 5,  7, 2;
  0, 1, 6,  8, 3;
  0, 1, 6, 10, 5;
  0, 1, 7, 12, 6, 1;
  ...

Links

Crossrefs

Row sums give A000009.
Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.

Formula

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).
T(n,k) = |A292047(n,k)| = |A292049(n,k)|.
Showing 1-4 of 4 results.

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

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