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 1-10 of 11 results. Next

A379599 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A088714.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 13, 0, 1, 4, 12, 32, 69, 0, 1, 5, 18, 58, 173, 419, 0, 1, 6, 25, 92, 321, 1054, 2809, 0, 1, 7, 33, 135, 523, 1971, 7039, 20353, 0, 1, 8, 42, 188, 790, 3248, 13158, 50632, 157199, 0, 1, 9, 52, 252, 1134, 4976, 21740, 94194, 387613, 1281993, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 27 2025

Keywords

Examples

			Square array begins:
  1,    1,    1,     1,     1,     1,     1, ...
  0,    1,    2,     3,     4,     5,     6, ...
  0,    3,    7,    12,    18,    25,    33, ...
  0,   13,   32,    58,    92,   135,   188, ...
  0,   69,  173,   321,   523,   790,  1134, ...
  0,  419, 1054,  1971,  3248,  4976,  7260, ...
  0, 2809, 7039, 13158, 21740, 33480, 49210, ...

Crossrefs

Columns k=0..1 give A000007, A088714.
Cf. A379168, A379598.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*a(n-j, j)));

Formula

See A088714.

A381566 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A087949.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 5, 0, 1, 5, 10, 13, 15, 16, 0, 1, 6, 15, 24, 33, 46, 59, 0, 1, 7, 21, 40, 63, 99, 164, 246, 0, 1, 8, 28, 62, 110, 188, 343, 662, 1131, 0, 1, 9, 36, 91, 180, 331, 638, 1344, 2961, 5655, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2025

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,    1,    1, ...
  0,  1,   2,   3,   4,    5,    6, ...
  0,  1,   3,   6,  10,   15,   21, ...
  0,  2,   6,  13,  24,   40,   62, ...
  0,  5,  15,  33,  63,  110,  180, ...
  0, 16,  46,  99, 188,  331,  552, ...
  0, 59, 164, 343, 638, 1110, 1845, ...

Crossrefs

Columns k=0..1 give A000007, A087949.
Cf. A379598, A381567, A381569.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, j)));

Formula

See A087949.

A384652 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384145.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 8, 0, 1, 4, 9, 20, 44, 0, 1, 5, 14, 37, 108, 298, 0, 1, 6, 20, 60, 198, 716, 2359, 0, 1, 7, 27, 90, 321, 1290, 5554, 21112, 0, 1, 8, 35, 128, 485, 2064, 9821, 48838, 209175, 0, 1, 9, 44, 175, 699, 3091, 15452, 84888, 476714, 2262121, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2025

Keywords

Examples

			Square array begins:
  1,    1,    1,    1,     1,     1,     1, ...
  0,    1,    2,    3,     4,     5,     6, ...
  0,    2,    5,    9,    14,    20,    27, ...
  0,    8,   20,   37,    60,    90,   128, ...
  0,   44,  108,  198,   321,   485,   699, ...
  0,  298,  716, 1290,  2064,  3091,  4434, ...
  0, 2359, 5554, 9821, 15452, 22805, 32315, ...

Crossrefs

Columns k=0..1 give A000007, A384145.
Cf. A379598, A384651, A384653, A384654.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-2*j+k, j)/(3*n-2*j+k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-2*j+k,j)/(3*n-2*j+k) * A(n-j,j).

A384653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384649.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 9, 0, 1, 4, 9, 22, 56, 0, 1, 5, 14, 40, 134, 432, 0, 1, 6, 20, 64, 240, 1012, 3935, 0, 1, 7, 27, 95, 381, 1779, 9039, 40820, 0, 1, 8, 35, 134, 565, 2780, 15596, 92246, 471633, 0, 1, 9, 44, 182, 801, 4071, 23950, 156597, 1051558, 5980210, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2025

Keywords

Examples

			Square array begins:
  1,    1,    1,     1,     1,     1,     1, ...
  0,    1,    2,     3,     4,     5,     6, ...
  0,    2,    5,     9,    14,    20,    27, ...
  0,    9,   22,    40,    64,    95,   134, ...
  0,   56,  134,   240,   381,   565,   801, ...
  0,  432, 1012,  1779,  2780,  4071,  5718, ...
  0, 3935, 9039, 15596, 23950, 34515, 47786, ...

Crossrefs

Columns k=0..1 give A000007, A384649.
Cf. A379598, A384651, A384652, A384654.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(4*n-3*j+k,j)/(4*n-3*j+k) * A(n-j,j).

A384654 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384650.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 10, 0, 1, 4, 9, 24, 69, 0, 1, 5, 14, 43, 162, 592, 0, 1, 6, 20, 68, 285, 1362, 6052, 0, 1, 7, 27, 100, 445, 2352, 13664, 70870, 0, 1, 8, 35, 140, 650, 3612, 23171, 157592, 928497, 0, 1, 9, 44, 189, 909, 5201, 34972, 263190, 2039543, 13404514, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2025

Keywords

Examples

			Square array begins:
  1,    1,     1,     1,     1,     1,     1, ...
  0,    1,     2,     3,     4,     5,     6, ...
  0,    2,     5,     9,    14,    20,    27, ...
  0,   10,    24,    43,    68,   100,   140, ...
  0,   69,   162,   285,   445,   650,   909, ...
  0,  592,  1362,  2352,  3612,  5201,  7188, ...
  0, 6052, 13664, 23171, 34972 ,49540, 67433, ...

Crossrefs

Columns k=0..1 give A000007, A384650.
Cf. A379598, A384651, A384652, A384653.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-4*j+k, j)/(5*n-4*j+k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(5*n-4*j+k,j)/(5*n-4*j+k) * A(n-j,j).

A381571 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381572.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 7, 0, 1, 6, 18, 38, 0, 1, 8, 33, 104, 267, 0, 1, 10, 52, 206, 735, 2232, 0, 1, 12, 75, 352, 1488, 6064, 21200, 0, 1, 14, 102, 550, 2626, 12246, 56510, 222556, 0, 1, 16, 133, 808, 4265, 21752, 112669, 581452, 2536661, 0, 1, 18, 168, 1134, 6537, 35812, 198808, 1140150, 6501267, 31010886, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2025

Keywords

Examples

			Square array begins:
  1,     1,     1,      1,      1,      1,      1, ...
  0,     2,     4,      6,      8,     10,     12, ...
  0,     7,    18,     33,     52,     75,    102, ...
  0,    38,   104,    206,    352,    550,    808, ...
  0,   267,   735,   1488,   2626,   4265,   6537, ...
  0,  2232,  6064,  12246,  21752,  35812,  55944, ...
  0, 21200, 56510, 112669, 198808, 327010, 512934, ...

Crossrefs

Columns k=0..1 give A000007, A381572.
Cf. A379598, A381573.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, 2*k*sum(j=0, n, binomial(2*n-j+2*k, j)/(2*n-j+2*k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = 2*k * Sum_{j=0..n} binomial(2*n-j+2*k,j)/(2*n-j+2*k) * A(n-j,j).

A381573 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381574.

Original entry on oeis.org

1, 1, 0, 1, 3, 0, 1, 6, 15, 0, 1, 9, 39, 118, 0, 1, 12, 72, 326, 1206, 0, 1, 15, 114, 651, 3345, 14712, 0, 1, 18, 165, 1120, 6822, 40200, 204385, 0, 1, 21, 225, 1760, 12123, 81675, 547146, 3143826, 0, 1, 24, 294, 2598, 19815, 145968, 1096080, 8239938, 52580328, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2025

Keywords

Examples

			Square array begins:
  1,     1,     1,     1,      1,      1, ...
  0,     3,     6,     9,     12,     15, ...
  0,    15,    39,    72,    114,    165, ...
  0,   118,   326,   651,   1120,   1760, ...
  0,  1206,  3345,  6822,  12123,  19815, ...
  0, 14712, 40200, 81675, 145968, 241773, ...

Crossrefs

Columns k=0..1 give A000007, A381574.
Cf. A379598, A381571.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, 3*k*sum(j=0, n, binomial(3*n-2*j+3*k, j)/(3*n-2*j+3*k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = 3*k * Sum_{j=0..n} binomial(3*n-2*j+3*k,j)/(3*n-2*j+3*k) * A(n-j,j).

A381602 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A120971.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 26, 0, 1, 4, 15, 60, 218, 0, 1, 5, 22, 103, 504, 2151, 0, 1, 6, 30, 156, 870, 4946, 23854, 0, 1, 7, 39, 220, 1329, 8511, 54430, 289555, 0, 1, 8, 49, 296, 1895, 12988, 93070, 655362, 3783568, 0, 1, 9, 60, 385, 2583, 18536, 141316, 1112382, 8496454, 52624689, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2025

Keywords

Examples

			Square array begins:
  1,     1,     1,     1,      1,      1,      1, ...
  0,     1,     2,     3,      4,      5,      6, ...
  0,     4,     9,    15,     22,     30,     39, ...
  0,    26,    60,   103,    156,    220,    296, ...
  0,   218,   504,   870,   1329,   1895,   2583, ...
  0,  2151,  4946,  8511,  12988,  18536,  25332, ...
  0, 23854, 54430, 93070, 141316, 200930, 273915, ...

Crossrefs

Columns k=0..1 give A000007, A120971, A120970(n+1).
Cf. A379598, A381603.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*a(n-j, 2*j)));

Formula

See A120971.

A381603 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A120973.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, 60, 0, 1, 4, 21, 132, 776, 0, 1, 5, 30, 217, 1708, 11802, 0, 1, 6, 40, 316, 2814, 25876, 201465, 0, 1, 7, 51, 430, 4113, 42510, 439446, 3759100, 0, 1, 8, 63, 560, 5625, 62016, 718647, 8155874, 75404151, 0, 1, 9, 76, 707, 7371, 84731, 1044228, 13270944, 162762498, 1608036861, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2025

Keywords

Examples

			Square array begins:
  1,      1,      1,      1,       1,       1,       1, ...
  0,      1,      2,      3,       4,       5,       6, ...
  0,      6,     13,     21,      30,      40,      51, ...
  0,     60,    132,    217,     316,     430,     560, ...
  0,    776,   1708,   2814,    4113,    5625,    7371, ...
  0,  11802,  25876,  42510,   62016,   84731,  111018, ...
  0, 201465, 439446, 718647, 1044228, 1421835, 1857631, ...

Crossrefs

Columns k=0..1 give A000007, A120973.
Cf. A379598, A381602.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n+k, j)/(3*n+k)*a(n-j, 3*j)));

Formula

See A120973.

A384651 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A162661.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 7, 0, 1, 4, 9, 18, 33, 0, 1, 5, 14, 34, 84, 189, 0, 1, 6, 20, 56, 159, 472, 1249, 0, 1, 7, 27, 85, 265, 882, 3057, 9237, 0, 1, 8, 35, 122, 410, 1460, 5615, 22190, 74972, 0, 1, 9, 44, 168, 603, 2256, 9166, 40053, 177149, 659042, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2025

Keywords

Examples

			Square array begins:
  1,    1,    1,    1,    1,     1,     1, ...
  0,    1,    2,    3,    4,     5,     6, ...
  0,    2,    5,    9,   14,    20,    27, ...
  0,    7,   18,   34,   56,    85,   122, ...
  0,   33,   84,  159,  265,   410,   603, ...
  0,  189,  472,  882, 1460,  2256,  3330, ...
  0, 1249, 3057, 5615, 9166, 14015, 20540, ...

Crossrefs

Columns k=0..1 give A000007, A162661.
Cf. A379598, A384652, A384653, A384654.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-j+k, j)/(2*n-j+k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(2*n-j+k,j)/(2*n-j+k) * A(n-j,j).
Showing 1-10 of 11 results. Next

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

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