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 1-4 of 4 results.

A336708 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} (-1)^(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, -1, 1, 1, 0, 0, 1, 1, 1, -1, 2, 1, 1, 2, 1, 0, -3, 1, 1, 3, 6, 1, 2, -1, 1, 1, 4, 14, 21, 1, 0, 11, 1, 1, 5, 25, 76, 80, 1, -5, -15, 1, 1, 6, 39, 182, 450, 322, 1, 0, -13, 1, 1, 7, 56, 355, 1447, 2818, 1347, 1, 14, 77, 1, 1, 8, 76, 611, 3532, 12175, 18352, 5798, 1, 0, -86
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, ...
  -1,  0, 1,   2,    3,     4,     5, ...
   0, -1, 1,   6,   14,    25,    39, ...
   2,  0, 1,  21,   76,   182,   355, ...
  -3,  2, 1,  80,  450,  1447,  3532, ...
  -1,  0, 1, 322, 2818, 12175, 37206, ...

Links

Crossrefs

Columns k=0-3 give: A007440, A090192, A000012, A106228.
Main diagonal gives A336713.
Cf. A336575, A336706, A336707, A336709.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(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+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 + x * A_k(x)).

Extensions

Typo in name corrected by Georg Fischer, Sep 19 2023

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)).

A335871 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

Original entry on oeis.org

1, 1, 3, 20, 234, 4159, 101538, 3182454, 122285201, 5575750271, 294529785168, 17697480642005, 1192398100081202, 89053864927236146, 7302988011333915878, 652439391227186881683, 63077327237347821501754, 6561701255914880362990927, 730833849063629052249986940
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Links

Crossrefs

Main diagonal of A336706.
Cf. A336578, A336712, A336713, A336714.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    {a(n) = if(n==0, 1, sum(k=1, n, binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}

A336707 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, 3, 6, 1, 1, 4, 11, 20, 1, 1, 5, 19, 45, 72, 1, 1, 6, 30, 100, 197, 272, 1, 1, 7, 44, 201, 562, 903, 1064, 1, 1, 8, 61, 364, 1445, 3304, 4279, 4272, 1, 1, 9, 81, 605, 3249, 10900, 20071, 20793, 17504, 1, 1, 10, 104, 940, 6502, 30526, 85128, 124996, 103049, 72896
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,   3,    4,     5,     6,     7,      8, ...
    6,  11,   19,    30,    44,    61,     81, ...
   20,  45,  100,   201,   364,   605,    940, ...
   72, 197,  562,  1445,  3249,  6502,  11857, ...
  272, 903, 3304, 10900, 30526, 73723, 158034, ...

Links

Crossrefs

Columns k=0-3 give: A071356(n-1), A001003, A007564, A118346.
Main diagonal gives A336712.
Cf. A336575, A336706, A336708, A336709.

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)).
Showing 1-4 of 4 results.

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