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.

A060099 G.f.: 1/((1-x^2)^3*(1-x)^4).

Original entry on oeis.org

1, 4, 13, 32, 71, 140, 259, 448, 742, 1176, 1806, 2688, 3906, 5544, 7722, 10560, 14223, 18876, 24739, 32032, 41041, 52052, 65429, 81536, 100828, 123760, 150892, 182784, 220116, 263568, 313956, 372096, 438957, 515508, 602889, 702240, 814891, 942172, 1085623
Offset: 0

Views

Author

Wolfdieter Lang, Apr 06 2001

Keywords

Comments

Fourth column (m=3) of triangle A060098.
Partial sums of A038163.
Equals the tetrahedral numbers, [1, 4, 10, 20, ...] convolved with the aerated triangular numbers, [1, 0, 3, 0, 6, 0, 10, ...]. [Gary W. Adamson, Jun 11 2009]

References

  • B. Broer, Hilbert series for modules of covariants, in Algebraic Groups and Their Generalizations..., Proc. Sympos. Pure Math., 56 (1994), Part I, 321-331. See p. 329.

Links

Crossrefs

Cf. A002620, A002624, A096338.
Cf. A001752 (for the similar series 1/((1-x)^4*(1-x^2))).
Cf. A028346 (for the similar series 1/((1-x)^4*(1-x^2)^2)).

Programs

  • Mathematica
    a[n_]:=If[OddQ[n],((1+n) (3+n) (5+n)^2 (7+n) (9+n))/5760,((2+n) (4+n) (6+n) (8+n) (15+10 n+n^2))/5760]; Map[a,Range[0,100]] (* Peter J. C. Moses, Mar 24 2013 *)
CoefficientList[Series[1/((1-x^2)^3*(1-x)^4),{x,0,100}],x] (* Peter J. C. Moses, Mar 24 2013 *)
LinearRecurrence[{4,-3,-8,14,0,-14,8,3,-4,1},{1,4,13,32,71,140,259,448,742,1176},40] (* Harvey P. Dale, Apr 06 2018 *)

Formula

a(n) = Sum_{} A060098(n+3, 3).
G.f.: 1/((1-x)^7*(1+x)^3).

A152205 Triangle read by rows, A000012 * A152204.

Original entry on oeis.org

1, 4, 9, 1, 16, 4, 25, 9, 1, 36, 16, 4, 49, 25, 9, 1, 64, 36, 16, 4, 81, 49, 25, 9, 1, 100, 64, 36, 16, 4, 121, 81, 49, 25, 9, 1, 144, 100, 64, 36, 16, 4, 169, 121, 81, 49, 25, 9, 1
Offset: 1

Views

Author

Gary W. Adamson, Nov 29 2008

Keywords

Comments

Row sums = A000292, the tetrahedral numbers.
From Gary W. Adamson, Feb 14 2010: (Start)
Let the triangle = M. Then lim_{n->inf} M^n = A173277 as a left-shifted vector: (1, 4, 13, 32, 74, 152, 298, ...) = A(x), where A(x) satisfies A000290 = A(x)/A(x^2), A000290 = integer squares.
M * [1, 2, 3, ...] = A001752: (1, 4, 11, 24, 46, 80, 130, ...).
M * [1, 3, 6, 10, ...] = A028346: (1, 4, 12, 28, 58, 108, ...). (End)

Examples

			First few rows of the triangle:
    1;
    4;
    9,   1;
   16,   4;
   25,   9,   1;
   36,  16,   4;
   49,  25,   9,   1;
   64,  36,  16,   4;
   81,  49,  25,   9,   1;
  100,  64,  36,  16,   4;
  121,  81,  49,  25,   9,   1;
  144, 100,  64,  36,  16,   4;
  169, 121,  81,  49,  25,   9,   1;
  ...

Crossrefs

Cf. A152204, A000292.
Cf. A173277, A001752, A028346. - Gary W. Adamson, Feb 14 2010

Formula

A000012 * A152204 = partial sums of A152204 by columns.

A062534 Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).

Original entry on oeis.org

1, -2, 1, 3, -1, 1, -4, 2, 0, 1, 5, -2, 2, 1, 1, -6, 3, 0, 3, 2, 1, 7, -3, 3, 3, 5, 3, 1, -8, 4, 0, 6, 8, 8, 4, 1, 9, -4, 4, 6, 14, 16, 12, 5, 1, -10, 5, 0, 10, 20, 30, 28, 17, 6, 1, 11, -5, 5, 10, 30, 50, 58, 45, 23, 7, 1, -12, 6, 0, 15, 40, 80, 108, 103, 68, 30, 8, 1, 13, -6, 6, 15, 55, 120, 188, 211, 171, 98, 38, 9, 1, -14, 7, 0, 21, 70, 175
Offset: 0

Views

Author

Henry Bottomley, Jun 25 2001

Keywords

Crossrefs

Rows are effectively (with minor adjustments): A038608, A001057, A027656, A008805, A006918, A002624, A028346. Cf. A058394 which (adjusting for signs and an overlap of three rows) is effectively the continuation of this table for negative n.

Formula

Each row is partial sum of preceding row, i.e. T(n, k)=T(n-1, k)+T(n, k-1) with T(0, k)=(k+1)*(-1)^k and T(n, 0)=1.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.