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 1-5 of 5 results.

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)).

A336713 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 6, 76, 1447, 37206, 1212194, 47975271, 2238595055, 120453255172, 7347494056729, 501273291296174, 37833413358907566, 3130557361463956074, 281854137496597897755, 27433898122963009937892, 2870816347095046227070383, 321430790732030793454519088
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2020

Keywords

Links

Crossrefs

Main diagonal of A336708.
Cf. A336578, A335871, A336712, A336714.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(-1)^(n - k) * 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, (-1)^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}

Formula

a(n) ~ exp(n - 1/2 - exp(-1)) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 04 2025

A336706 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} 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, 2, 2, 1, 1, 3, 5, 4, 1, 1, 4, 11, 14, 9, 1, 1, 5, 20, 45, 42, 21, 1, 1, 6, 32, 113, 197, 132, 51, 1, 1, 7, 47, 234, 688, 903, 429, 127, 1, 1, 8, 65, 424, 1854, 4404, 4279, 1430, 323, 1, 1, 9, 86, 699, 4159, 15490, 29219, 20793, 4862, 835
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,   2,   3,    4,     5,     6,      7, ...
   2,   5,  11,   20,    32,    47,     65, ...
   4,  14,  45,  113,   234,   424,    699, ...
   9,  42, 197,  688,  1854,  4159,   8192, ...
  21, 132, 903, 4404, 15490, 43097, 101538, ...

Links

Crossrefs

Columns k=0-3 give: A001006(n-1), A000108, A001003, A108447.
Main diagonal gives A335871.
Cf. A336575, A336707, A336708, A336709.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[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, 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)).

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)).

A336727 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} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 0, 1, 1, 1, -3, 1, 5, 2, 1, 1, 1, -4, 5, 10, -3, 0, 1, 1, 1, -5, 11, 9, -38, -21, -5, 1, 1, 1, -6, 19, -4, -103, 28, 51, 0, 1, 1, 1, -7, 29, -35, -174, 357, 289, 41, 14, 1, 1, 1, -8, 41, -90, -203, 1176, -131, -1262, -391, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2020

Keywords

Examples

			  1,  1,   1,   1,    1,    1,    1, ...
  1,  1,   1,   1,    1,    1,    1, ...
  1,  0,  -1,  -2,   -3,   -4,   -5, ...
  1, -1,  -1,   1,    5,   11,   19, ...
  1,  0,   5,  10,    9,   -4,  -35, ...
  1,  2,  -3, -38, -103, -174, -203, ...
  1,  0, -21,  28,  357, 1176, 2575, ...

Links

Crossrefs

Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.
Main diagonal gives A336728.
Cf. A243631, A307884, A336708, A336709.

Programs

  • Mathematica
    T[0, k_] := 1; T[n_, k_] := Sum[If[k == 0, Boole[n == j],(-k)^(n - j)] * Binomial[n, j] * Binomial[n , j - 1], {j, 1, n}] / n; Table[T[k, n- k], {n, 0, 11}, {k, 0, n}] //Flatten (* Amiram Eldar, Aug 02 2020 *)
  • PARI
    {T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}
  • PARI
    {T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+k*x*A)); polcoef(A, n)}
  • PARI
    {T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}

Formula

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x) / (1 + k * x * A_k(x)).
A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).
T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).
(n+1) * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-2) * T(n-2,k) for n>1. - Seiichi Manyama, Aug 08 2020
Showing 1-5 of 5 results.

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