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.

A378289 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n+r+k,r) * binomial(r,n-r)/(n+r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 26, 38, 0, 1, 5, 18, 49, 105, 154, 0, 1, 6, 25, 80, 210, 444, 654, 0, 1, 7, 33, 120, 363, 927, 1944, 2871, 0, 1, 8, 42, 170, 575, 1672, 4191, 8734, 12925, 0, 1, 9, 52, 231, 858, 2761, 7810, 19305, 40040, 59345, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 21 2024

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,  10,   26,   49,   80,   120,   170, ...
  0,  38,  105,  210,  363,   575,   858, ...
  0, 154,  444,  927, 1672,  2761,  4290, ...
  0, 654, 1944, 4191, 7810, 13325, 21385, ...

Crossrefs

Columns k=0..3 give A000007, A001002, A052706(n+2), A052703(n+3).
Cf. A009766, A026300, A378290, A378291, A378292.

Programs

  • PARI
    T(n, k, t=2, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A001002.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-2,k+2) for n > 1.

A378291 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3*n-2*r+k,r) * binomial(r,n-r)/(3*n-2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 6, 0, 1, 4, 9, 16, 20, 0, 1, 5, 14, 31, 56, 72, 0, 1, 6, 20, 52, 114, 208, 273, 0, 1, 7, 27, 80, 201, 438, 806, 1073, 0, 1, 8, 35, 116, 325, 800, 1739, 3220, 4333, 0, 1, 9, 44, 161, 495, 1341, 3260, 7077, 13168, 17869, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 21 2024

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,   6,  16,   31,   52,   80,  116, ...
  0,  20,  56,  114,  201,  325,  495, ...
  0,  72, 208,  438,  800, 1341, 2118, ...
  0, 273, 806, 1739, 3260, 5615, 9119, ...

Crossrefs

Columns k=0..1 give A000007, A186996.
Cf. A009766, A026300, A378289, A378290, A378292.
Cf. A378237.

Programs

  • PARI
    T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + x * A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A186996.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-2,k+3) for n > 1.

A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2*n+k,r) * binomial(r,n-r)/(2*n+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 0, 1, 4, 12, 28, 46, 0, 1, 5, 18, 52, 123, 207, 0, 1, 6, 25, 84, 240, 572, 979, 0, 1, 7, 33, 125, 407, 1155, 2769, 4797, 0, 1, 8, 42, 176, 635, 2028, 5733, 13806, 24138, 0, 1, 9, 52, 238, 936, 3276, 10332, 29136, 70414, 123998, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 21 2024

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,  11,   28,   52,    84,   125,   176, ...
  0,  46,  123,  240,   407,   635,   936, ...
  0, 207,  572, 1155,  2028,  3276,  4998, ...
  0, 979, 2769, 5733, 10332, 17140, 26860, ...

Crossrefs

Columns k=0..2 give A000007, A006605, A143927.
Cf. A009766, A026300, A378289, A378290, A378291.

Programs

  • PARI
    T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A006605.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-2,k+3) for n > 1.
Showing 1-3 of 3 results.

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