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.

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A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Examples

			Square array begins:
    1,   1,  1,  1,   1,    1,     1, ...
    1,   1,  1,  1,   1,    1,     1, ...
   -2,  -1,  0,  1,   2,    3,     4, ...
    2,  -1, -1,  2,   8,   17,    29, ...
    4,   5,  0,  5,  36,  109,   240, ...
  -24,  -3,  2, 13, 177,  766,  2177, ...
   48, -21,  0, 36, 922, 5699, 20910, ...

Links

Crossrefs

Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.
Main diagonal gives A336714.
Cf. A336575, A336706, A336707, A336708.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, (-2)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
  • PARI
    {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+2*x*A)); polcoef(A, n)}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + 2 * x * A_k(x)).

A307968 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 + k*x + sqrt(1 + 2*k*x + k*(k+4)*x^2)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 2, 0, 1, -4, 6, 4, -3, 0, 1, -5, 12, 0, -24, -1, 0, 1, -6, 20, -16, -63, 48, 11, 0, 1, -7, 30, -50, -96, 297, 24, -15, 0, 1, -8, 42, -108, -75, 896, -621, -464, -13, 0, 1, -9, 56, -196, 72, 1875, -3904, -1053, 1376, 77, 0
Offset: 0

Views

Author

Seiichi Manyama, May 08 2019

Keywords

Examples

			Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,  -1,   -2,    -3,    -4,     -5,     -6, ...
   0,   0,    2,     6,    12,     20,     30, ...
   0,   2,    4,     0,   -16,    -50,   -108, ...
   0,  -3,  -24,   -63,   -96,    -75,     72, ...
   0,  -1,   48,   297,   896,   1875,   3024, ...
   0,  11,   24,  -621, -3904, -13125, -32184, ...
   0, -15, -464, -1053,  6912,  53125, 200880, ...

Links

Crossrefs

Columns k=0..2 give A000007, A007440(n+1), A307969.
Main diagonal gives A307946.
Cf. A107267, A307819.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

Formula

A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 - k*x - k*x^2)^(n+1).
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n,2*j) * A000108(j).
(n+2) * A(n,k) = -k * (2*n+1) * A(n-1,k) - k * (k+4) * (n-1) * A(n-2,k).
Showing 1-2 of 2 results.

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