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.

A307883 Square array read by descending antidiagonals: T(n, k) where column k is the expansion of 1/sqrt(1 - 2*(k+1)*x + ((k-1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 20, 1, 1, 5, 22, 63, 70, 1, 1, 6, 33, 136, 321, 252, 1, 1, 7, 46, 245, 886, 1683, 924, 1, 1, 8, 61, 396, 1921, 5944, 8989, 3432, 1, 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1, 1, 10, 97, 848, 6145, 33876, 127905, 281488, 265729, 48620, 1
Offset: 0

Views

Author

Seiichi Manyama, May 02 2019

Keywords

Comments

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) - k*x*y). - Seiichi Manyama, Jul 11 2020
More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 - (k+r-1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

Examples

			Square array begins:
  1,   1,    1,     1,      1,      1,      1, ...
  1,   2,    3,     4,      5,      6,      7, ...
  1,   6,   13,    22,     33,     46,     61, ...
  1,  20,   63,   136,    245,    396,    595, ...
  1,  70,  321,   886,   1921,   3606,   6145, ...
  1, 252, 1683,  5944,  15525,  33876,  65527, ...
  1, 924, 8989, 40636, 127905, 324556, 712909, ...
Seen as a triangle T(n, k):
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  1;
  [3] 1, 3,  6,   1;
  [4] 1, 4, 13,  20,    1;
  [5] 1, 5, 22,  63,   70,     1;
  [6] 1, 6, 33, 136,  321,   252,     1;
  [7] 1, 7, 46, 245,  886,  1683,   924,     1;
  [8] 1, 8, 61, 396, 1921,  5944,  8989,  3432,     1;
  [9] 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1;

Links

Crossrefs

Columns k=0..6 give A000012, A000984, A001850, A069835, A084771, A084772, A098659.
Main diagonal gives A187021.
T(n,n+1) gives A335309.
Cf. A307855, A307884, A307910.

Programs

  • Maple
    # Seen as a triangle read by rows:
T := (n, k) -> simplify(hypergeom([-k, -k], [1], n - k)):
seq(lprint(seq(T(n, k), k = 0..n)), n = 0..9);  # Peter Luschny, May 13 2024
  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)
(* Seen as a triangle read by rows: *)
T[n_, k_] := HypergeometricPFQ[{-k, -k}, {1}, n - k];
Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, May 13 2024 *)

Formula

T(n,k) is the coefficient of x^n in the expansion of (1 + (k+1)*x + k*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (k+1) * (2*n-1) * T(n-1,k) - (k-1)^2 * (n-1) * T(n-2,k).
T(n,k) = hypergeom([-k, -k], [1], n - k), (triangular form). - Detlef Meya, May 13 2024

A307819 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*k*x + k*(k+4)*x^2).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, 0, 5, 0, 1, -4, 3, 16, -5, 0, 1, -5, 8, 27, -56, -11, 0, 1, -6, 15, 32, -189, 48, 41, 0, 1, -7, 24, 25, -416, 567, 384, -29, 0, 1, -8, 35, 0, -725, 2176, 189, -1920, -125, 0, 1, -9, 48, -49, -1080, 5625, -4864, -11259, 3168, 365, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2019

Keywords

Examples

			Square array begins:
   1,   1,     1,      1,      1,      1,      1, ...
   0,  -1,    -2,     -3,     -4,     -5,     -6, ...
   0,  -1,     0,      3,      8,     15,     24, ...
   0,   5,    16,     27,     32,     25,      0, ...
   0, -11,    48,    567,   2176,   5625,  11664, ...
   0,  41,   384,    189,  -4864, -24375, -74304, ...
   0, -29, -1920, -11259, -23552,   9375, 228096, ...

Links

Crossrefs

Columns k=0..3 give A000007, (-1)^n * A098331, A116093, (-1)^n * A098340.
Main diagonal gives A307911.
Cf. A307860, A307884, A307910.

Programs

  • Mathematica
    A[n_, k_] := (-k)^n*Hypergeometric2F1[(1-n)/2, -n/2, 1, -4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

Formula

A(n,k) is the coefficient of x^n in the expansion of (1 - k*x - k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = -k * (2*n-1) * A(n-1,k) - k * (k+4) * (n-1) * A(n-2,k).

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1
Offset: 0

Views

Author

Seiichi Manyama, May 01 2019

Keywords

Examples

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

Links

Crossrefs

Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

Formula

A(n,k) is the coefficient of x^n in the expansion of (1 + x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * binomial(2*j,j).
D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).
Showing 1-3 of 3 results.

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