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 1-6 of 6 results.

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, 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, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 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,   0,  -1,   -2,   -3,   -4,     -5, ...
  1,  -2,  -3,   -2,    1,    6,     13, ...
  1,   0,  11,   28,   45,   56,     55, ...
  1,   6,   1,  -74, -255, -554,   -959, ...
  1,   0, -81,  -92,  477, 2376,   6475, ...
  1, -20, 141, 1324, 2689, -804, -20195, ...

Links

Crossrefs

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.
Main diagonal gives A307885.
T(n,n-1) gives A335310.
Cf. A307883, A336179.

Programs

  • 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 *)

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).

A336180 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^3.

Original entry on oeis.org

1, 0, -11, 136, -639, -25624, 1133245, -27431424, 259448833, 17402599792, -1405909697499, 63884679938960, -1830503703899519, -5324845289379264, 5494299851213052685, -496909924804074650624, 30201149245542631276545, -1236819213672144144878752, 5410434345252588202534741
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2020

Keywords

Links

Crossrefs

Main diagonal of A336179.
Cf. A241247, A307885.

Programs

  • Maple
    a := n -> hypergeom([-n, -n, -n], [1, 1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020
  • Mathematica
    Array[Function[n, 1 + Sum[(-n)^k Binomial[n, k]^3, {k, n}]], 19, 0] (* Jan Mangaldan, Jul 14 2020 *)
  • PARI
    {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^3)}

Formula

a(n) = hypergeom([-n, -n, -n], [1, 1], n). - Peter Luschny, Dec 22 2020

A336182 a(n) = Sum_{k=0..n} (-3)^k * binomial(n,k)^3.

Original entry on oeis.org

1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372, 365951356, -1601617712, -19715085584, 233866581856, 285409397056, -20406741254144, 90043530872218, 1169513126877676, -13961261999882204, -18779832792734384, 1270510266589738636, -5584024444211882792
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2020

Keywords

Comments

Diagonal of the rational function 1 / (1 + y + z + x*y + y*z - 3*z*x - 2*x*y*z).
Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) + 3*x*y*z).

Links

Crossrefs

Column k=3 of A336179.
Cf. A206180.

Programs

  • Maple
    f:= gfun:-rectoproc({(24*n^3 + 176*n^2 + 416*n + 320)*a(n + 1) + (279*n^3 + 2325*n^2 + 6382*n + 5776)*a(n + 2) + (18*n^3 + 168*n^2 + 514*n + 512)*a(n + 3) + (3*n^3 + 31*n^2 + 104*n + 112)*a(n + 4), a(0) = 1, a(1) = -2, a(2) = -14, a(3) = 136},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 12 2020
  • Mathematica
    a[n_] := Sum[(-3)^k * Binomial[n, k]^3, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jul 11 2020 *)
  • PARI
    {a(n) = sum(k=0, n, (-3)^k*binomial(n,k)^3)}

Formula

From Robert Israel, Jul 12 2020: (Start)
a(n) = hypergeom([-n,-n,-n],[1,1],3).
(24*n^3 + 176*n^2 + 416*n + 320)*a(n + 1) + (279*n^3 + 2325*n^2 + 6382*n + 5776)*a(n + 2) + (18*n^3 + 168*n^2 + 514*n + 512)*a(n + 3) + (3*n^3 + 31*n^2 + 104*n + 112)*a(n + 4)=0. (End)

A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 10, 1, 1, 4, 21, 56, 1, 1, 5, 34, 171, 346, 1, 1, 6, 49, 352, 1521, 2252, 1, 1, 7, 66, 605, 3946, 14283, 15184, 1, 1, 8, 85, 936, 8065, 46744, 138909, 104960, 1, 1, 9, 106, 1351, 14346, 113525, 573616, 1385163, 739162, 1, 1, 10, 129, 1856, 23281, 231876, 1656145, 7217536, 14072193, 5280932, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2020

Keywords

Comments

Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z).
Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z).

Examples

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,   10,    21,    34,     49,     66, ...
  1,   56,   171,   352,    605,    936, ...
  1,  346,  1521,  3946,   8065,  14346, ...
  1, 2252, 14283, 46744, 113525, 231876, ...

Crossrefs

Columns k=0-6 give: A000012, A000172, A206178, A206180, A216483, A216636, A216698.
Main diagonal gives A241247.
Cf. A307883, A336179, A336187.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[k^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)

A336181 a(n) = Sum_{k=0..n} (-2)^k * binomial(n,k)^3.

Original entry on oeis.org

1, -1, -11, 47, 241, -2281, -3779, 104831, -110207, -4415281, 16955269, 161498831, -1252782959, -4376471737, 73606867309, 11876256767, -3715460133887, 9838677757343, 160921055160469, -957644184613393, -5553047963695439, 65231925087461879, 102267746634093469
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2020

Keywords

Comments

Diagonal of the rational function 1 / (1 + y + z + x*y + y*z - 2*z*x - x*y*z).
Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) + 2*x*y*z).

Crossrefs

Column k=2 of A336179.
Cf. A206178.

Programs

  • Mathematica
    a[n_] := Sum[(-2)^k * Binomial[n, k]^3, {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Jul 11 2020 *)
  • PARI
    {a(n) = sum(k=0, n, (-2)^k*binomial(n, k)^3)}

A336201 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, -3, 0, 1, 1, -3, -14, 11, 0, 1, 1, -4, -47, 136, 1, 0, 1, 1, -5, -134, 909, 106, -81, 0, 1, 1, -6, -347, 4736, 3585, -8492, 141, 0, 1, 1, -7, -846, 21655, 61906, -323523, 35344, 363, 0, 1, 1, -8, -1983, 91512, 771601, -8065624, 2201809, 395008, -1791, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2020

Keywords

Comments

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) + k * Product_{j=1..k} x_j) for k>0.

Examples

			Square array begins:
  1, 1,   1,     1,       1,        1, ...
  1, 0,  -1,    -2,      -3,       -4, ...
  1, 0,  -3,   -14,     -47,     -134, ...
  1, 0,  11,   136,     909,     4736, ...
  1, 0,   1,   106,    3585,    61906, ...
  1, 0, -81, -8492, -323523, -8065624, ...

Crossrefs

Columns k=0-3 give: A000012, A000007, (-1)^n*A098332(n), A336182.
Main diagonal gives A336202.
Cf. A309010, A336179, A336187.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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