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.

A362078 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 22, 35, 1, 1, 9, 37, 105, 126, 1, 1, 11, 55, 215, 511, 462, 1, 1, 13, 76, 369, 1271, 2534, 1716, 1, 1, 15, 100, 571, 2526, 7651, 12720, 6435, 1, 1, 17, 127, 825, 4401, 17577, 46614, 64449, 24310, 1, 1, 19, 157, 1135, 7026, 34412, 123810, 286599, 328900, 92378
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Examples

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    1,   1,    1,    1,    1,    1, ...
    3,   5,    7,    9,   11,   13, ...
   10,  22,   37,   55,   76,  100, ...
   35, 105,  215,  369,  571,  825, ...
  126, 511, 1271, 2526, 4401, 7026, ...

Crossrefs

Columns k=0..3 give A088218, A213684, A362087, A362088.
Main diagonal gives A362080.
Cf. A362079.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(n+j-1, j)*binomial(k*j, n-j));

Formula

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-n,j) * binomial(k*j,n-j) = Sum_{j=0..n} binomial(n+j-1,j) * binomial(k*j,n-j).

A362079 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^n)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 28, 45, 0, 1, 5, 18, 55, 145, 251, 0, 1, 6, 25, 92, 315, 896, 1624, 0, 1, 7, 33, 140, 571, 2106, 6328, 11908, 0, 1, 8, 42, 200, 930, 4076, 15946, 50212, 97545, 0, 1, 9, 52, 273, 1410, 7026, 32718, 134730, 441489, 880660, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Examples

			Square array begins:
  1,   1,   1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5, ...
  0,   3,   7,   12,   18,   25, ...
  0,  10,  28,   55,   92,  140, ...
  0,  45, 145,  315,  571,  930, ...
  0, 251, 896, 2106, 4076, 7026, ...

Crossrefs

Columns k=0..3 give A000007, A099237, A362084, A362085.
Main diagonal gives A362080.
Cf. A362078.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(n*j, n-j));

Formula

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(n*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(n*j,n-j).

A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 47, 8, 0, 1, 6, 40, 124, 198, 118, 13, 0, 1, 7, 57, 235, 571, 681, 290, 21, 0, 1, 8, 77, 398, 1320, 2500, 2263, 702, 34, 0, 1, 9, 100, 623, 2640, 7026, 10504, 7341, 1677, 55, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Examples

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   7,  15,   26,   40, ...
  0, 3,  18,  55,  124,  235, ...
  0, 5,  47, 198,  571, 1320, ...
  0, 8, 118, 681, 2500, 7026, ...

Crossrefs

Columns k=0..3 give A000007, A000045(n+1), A362126, A382614.
Main diagonal gives A362080.
Cf. A362078, A362079.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(k*j, n-j));

Formula

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(k*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(k*j,n-j).
Showing 1-3 of 3 results.

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

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