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 1-3 of 3 results.

A185292 Expansion of (x*(1+x)/(1-x^3))^4.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 6, 8, 17, 24, 26, 44, 60, 60, 90, 120, 115, 160, 210, 196, 259, 336, 308, 392, 504, 456, 564, 720, 645, 780, 990, 880, 1045, 1320, 1166, 1364, 1716, 1508, 1742, 2184, 1911, 2184, 2730, 2380, 2695
Offset: 0

Views

Author

Philippe Deléham, Jan 25 2012

Keywords

Comments

Expansion of ((x+x^2)/(1-x^3))^k for k = 4 ; for k=1 see A011655, for k = 2 see A186731, for k = 3 see A185395.
Column k = 4 of triangle in A198295.

Links

Crossrefs

Cf. A011655, A186731, A185395, A198295

Programs

  • Mathematica
    CoefficientList[Series[(x*(1 + x)/(1 - x^3))^4, {x, 0, 50}], x] (* G. C. Greubel, Jun 25 2017 *)
  • PARI
    x='x+O('x^50); concat([0,0,0,0], Vec((x*(1+x)/(1-x^3))^4)) \\ G. C. Greubel, Jun 25 2017

Formula

G.f.: (x*(1+x)/(1-x^3))^4.

A185395 a(3n) = n^2, a(3n+1) = a(3n+2) = 3*n*(n+1)/2.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 4, 9, 9, 9, 18, 18, 16, 30, 30, 25, 45, 45, 36, 63, 63, 49, 84, 84, 64, 108, 108, 81, 135, 135, 100, 165, 165, 121, 198, 198, 144, 234, 234, 169, 273, 273, 196, 315, 315, 225, 360, 360, 256
Offset: 0

Views

Author

Philippe Deléham, Jan 21 2012

Keywords

Comments

Expansion of ((x+x^2)/(1-x^3))^k with k = 3 ; for k = 1 see A011655, for k = 2 see A186731, for k = 4 see A185292.

Links

Crossrefs

Column k = 3 of triangle in A198295.
Cf. A000217, A000290, A011655, A045943, A072691, A186731, A185292.

Programs

  • Mathematica
    LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{0,0,0,1,3,3,4,9,9},50] (* Harvey P. Dale, Jan 23 2013 *)
  • PARI
    x='x+O('x^50); concat([0, 0, 0], Vec((x*(1+x)/(1-x^3))^3)) \\ G. C. Greubel, Jun 29 2017

Formula

G.f.: (x*(1+x)/(1-x^3))^3.
From Amiram Eldar, May 10 2025: (Start)
Sum_{n>=3} 1/a(n) = Pi^2/6 + 4/3.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). (End)

A198295 Riordan array (1, x*(1+x)/(1-x^3)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 3, 1, 0, 1, 2, 3, 4, 1, 0, 0, 4, 4, 6, 5, 1, 0, 1, 2, 9, 8, 10, 6, 1, 0, 1, 3, 9, 17, 15, 15, 7, 1, 0, 0, 6, 9, 24, 30, 26, 21, 8, 1, 0, 1, 3, 18, 26, 51, 51, 42, 28, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 26 2012

Keywords

Comments

Triangle T(n,k), read by rows, given by (0, 1, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Antidiagonals sums: see A159284.

Examples

			Triangle begins:
1
0, 1
0, 1, 1
0, 0, 2, 1
0, 1, 1, 3, 1
0, 1, 2, 3, 4, 1
0, 0, 4, 4, 6, 5, 1
0, 1, 2, 9, 8, 10, 6, 1
0, 1, 3, 9, 17, 15, 15, 7, 1

References

  • A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.

Links

Crossrefs

Cf. Columns: A000007, A011655, A186731, A185395, A185292.
Cf. Diagonals: A000012, A001477, A161680, A000125.

Formula

Sum_{k, 0<=k<=n} T(n,k) = A001590(n+2), n>0.
Sum_{k, 0<=k<=n}T(n,k)*(-1)^(n-k) = A078056(n-1), n>0.
T(n,n) = A000012(n), T(n+1,n) = A001477(n) = n, T(n+2,n) = A161680(n) = A000217(n-1); T(n+3,n) = A000125(n-1), n>=1.
G.f.: (-1+x)*(1+x+x^2)/(-1+x^3+x*y+x^2*y). - R. J. Mathar, Aug 11 2015
Showing 1-3 of 3 results.

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