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.

A319753 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - k*x^j).

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 1, 3, 8, 6, 0, 1, 1, 4, 15, 24, 14, 0, 0, 1, 5, 24, 60, 78, 27, 0, 1, 1, 6, 35, 120, 252, 232, 60, 0, 0, 1, 7, 48, 210, 620, 1005, 720, 117, 0, 0, 1, 8, 63, 336, 1290, 3096, 4080, 2152, 246, 0, 0, 1, 9, 80, 504, 2394, 7735, 15600, 16305, 6528, 490, 0, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

Examples

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
  -1,  0,   1,    2,     3,     4,  ...
  -1,  0,   3,    8,    15,    24,  ...
   0,  0,   6,   24,    60,   120,  ...
   0,  0,  14,   78,   252,   620,  ...
   1,  0,  27,  232,  1005,  3096,  ...
		

Crossrefs

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[(1 - x^i)/(1 - k x^i), {i, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
    Table[Function[k, SeriesCoefficient[Exp[Sum[Sum[d (k^(i/d) - 1), {d, Divisors[i]}] x^i/i, {i, n}]], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=1} (1 - x^j)/(1 - k*x^j).
G.f. of column k: exp(Sum_{j>=1} ( Sum_{d|j} d*(k^(j/d) - 1) ) * x^j/j).