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.

A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -2, 0, 0, 1, -4, -3, 2, 0, 0, 1, -5, -4, 6, 2, 1, 0, 1, -6, -5, 12, 6, 6, 0, 0, 1, -7, -6, 20, 12, 15, -2, 1, 0, 1, -8, -7, 30, 20, 28, -12, 2, 0, 0, 1, -9, -8, 42, 30, 45, -36, -3, -6, 0, 0, 1, -10, -9, 56, 42, 66, -80
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0,  0,  2,  6, 12, ...
   0,  0,  2,  6, 12, ...

Links

Crossrefs

Columns k=0..3 give A000007, A010815, A070877, A292128.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292132.
Cf. A246935, A286957.

A298986 a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k.

Original entry on oeis.org

1, -1, -4, 9, 48, 100, -756, -3479, -1600, 24462, 225900, 364573, -643536, -9251736, -36989316, -32397975, 165039872, 1725828525, 5338814652, 8082713829, -26321848400, -233434232766, -811526778964, -1731126953532, 1151302859712, 23632432765000, 113461901639788, 287935019845749
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 31 2018

Keywords

Crossrefs

Cf. A073592, A266964, A292132, A297324, A298985, A298987, A298988.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

A303189 a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).

Original entry on oeis.org

1, -1, -1, 5, 7, 21, -94, -117, -404, -840, 3541, 4536, 14412, 31313, 72175, -249424, -262828, -930639, -1895460, -4441316, -8085972, 24112570, 26214408, 87131883, 180197979, 411759028, 748154122, 1525043990, -3554837744, -3210408245, -11955482059, -23817949142, -55221348072
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Examples

			a(0) = 1;
a(1) = [x^1] (1 - x) = -1;
a(2) = [x^2] (1 - 2*x)*(1 - x^2) = -1;
a(3) = [x^3] (1 - 3*x)*(1 - 2*x^2)*(1 - x^3) = 5;
a(4) = [x^4] (1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4) = 7;
a(5) = [x^5] (1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5) = 21, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - (n - k + 1)*x^k) begins:
n = 0: (1),  0,   0,   0,   0,   0,  ...
n = 1:  1, (-1),  0,   0,   0,   0,  ...
n = 2:  1,  -2, (-1),  2,   0,   0   ...
n = 3:  1,  -3,  -2,  (5),  3,   2,  ...
n = 4:  1,  -4,  -3,  10,  (7), 10,  ...
n = 5:  1,  -5,  -4,  17,  13, (21), ...

Crossrefs

Cf. A022661, A292132, A303173, A303175, A303188, A303190.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 32}]
Showing 1-3 of 3 results.

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