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 1-8 of 8 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

A307885 Coefficient of x^n in (1 - (n-1)*x - n*x^2)^n.

Original entry on oeis.org

1, 0, -3, 28, -255, 2376, -20195, 71688, 3834369, -187855280, 6676401501, -220595216280, 7180102389889, -234023553073296, 7631745228481725, -245429882267144624, 7501602903392006145, -196609711096827812448, 2542435002501531333949
Offset: 0

Views

Author

Seiichi Manyama, May 02 2019

Keywords

Comments

Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*(n-1)*x + ((n+1)*x)^2).

Links

Crossrefs

Main diagonal of A307884.
Cf. A187021.

Programs

  • Maple
    A307885:= n -> simplify(hypergeom([-n,-n], [1], -n));
seq(A307885(n), n = 0..30); # G. C. Greubel, May 31 2020
  • Mathematica
    Table[Hypergeometric2F1[-n, -n, 1, -n], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2019 *)
  • PARI
    {a(n) = polcoef((1-(n-1)*x-n*x^2)^n, n)}
  • PARI
    {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^2)}
  • PARI
    {a(n) = sum(k=0, n, (-n-1)^(n-k)*binomial(n, k)*binomial(n+k, k))}
  • Sage
    [ hypergeometric([-n, -n], [1], -n).simplify_hypergeometric() for n in (0..30)] # G. C. Greubel, May 31 2020

Formula

a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^2.
a(n) = Sum_{k=0..n} (-n-1)^(n-k) * binomial(n,k) * binomial(n+k,k).
a(n) = Hypergeometric2F1(-n, -n, 1, -n). - Vaclav Kotesovec, May 07 2019
a(n) = n! * [x^n] exp((1 - n)*x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020

A335333 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*(2*k+1)*x + x^2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 13, 1, 1, 7, 37, 63, 1, 1, 9, 73, 305, 321, 1, 1, 11, 121, 847, 2641, 1683, 1, 1, 13, 181, 1809, 10321, 23525, 8989, 1, 1, 15, 253, 3311, 28401, 129367, 213445, 48639, 1, 1, 17, 337, 5473, 63601, 458649, 1651609, 1961825, 265729, 1
Offset: 0

Views

Author

Seiichi Manyama, Jun 02 2020

Keywords

Examples

			Square array begins:
  1,    1,     1,      1,      1,       1, ...
  1,    3,     5,      7,      9,      11, ...
  1,   13,    37,     73,    121,     181, ...
  1,   63,   305,    847,   1809,    3311, ...
  1,  321,  2641,  10321,  28401,   63601, ...
  1, 1683, 23525, 129367, 458649, 1256651, ...

Links

Crossrefs

Columns k=0..4 give A000012, A001850, A006442, A084768, A084769.
Rows n=0..6 give A000012, A005408, A003154(n+1), A160674, A144124, A335338, A144126.
Main diagonal gives A331656.
T(n,n-1) gives A331657.
Cf. A307883, A307884.

Programs

  • Mathematica
    T[n_, k_] := LegendreP[n, 2*k + 1]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)
  • PARI
    T(n, k) = pollegendre(n, 2*k+1);

Formula

T(n,k) is the coefficient of x^n in the expansion of (1 + (2*k+1)*x + k*(k+1)*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * (k+1)^(n-j) * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (n-1) * T(n-2,k).
T(n,k) = P_n(2*k+1), where P_n is n-th Legendre polynomial.
From Seiichi Manyama, Aug 30 2025: (Start)
T(n,k) = (-1)^n * Sum_{j=0..n} (1/(2*(2*k+1)))^(n-2*j) * binomial(-1/2,j) * binomial(j,n-j).
T(n,k) = Sum_{j=0..floor(n/2)} (k*(k+1))^j * (2*k+1)^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
E.g.f. of column k: exp((2*k+1)*x) * BesselI(0, 2*sqrt(k*(k+1))*x). (End)

A336179 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, 0, 1, 1, -1, -6, 1, 1, -2, -11, 0, 1, 1, -3, -14, 47, 90, 1, 1, -4, -15, 136, 241, 0, 1, 1, -5, -14, 261, 106, -2281, -1680, 1, 1, -6, -11, 416, -639, -8492, -3779, 0, 1, 1, -7, -6, 595, -2294, -17523, 35344, 104831, 34650, 1, 1, -8, 1, 792, -5135, -25624, 188049, 395008, -110207, 0, 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,  0,    -1,    -2,     -3,     -4, ...
  1, -6,   -11,   -14,    -15,    -14, ...
  1,  0,    47,   136,    261,    416, ...
  1, 90,   241,   106,   -639,  -2294, ...
  1,  0, -2281, -8492, -17523, -25624, ...

Links

Crossrefs

Columns k=0-3 give: A000012, A245086, A336181, A336182.
Main diagonal gives A336180.
Cf. A307884, A336163.

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 *)

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).

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

A335310 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * (-n)^(n-k).

Original entry on oeis.org

1, 1, -2, 11, -74, 477, -804, -84425, 3315334, -102211207, 3005297956, -88338323709, 2627003399164, -78764141488043, 2341929797646648, -66394419743289105, 1609460569459689286, -18001777147777896975, -1625299659961386724524, 196005371138608184827003
Offset: 0

Views

Author

Ilya Gutkovskiy, May 31 2020

Keywords

Links

Crossrefs

Cf. A098332, A116091, A126869, A307884, A307885, A331657, A335309.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] (-n)^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[SeriesCoefficient[1/Sqrt[1 + 2 (n - 2) x + n^2 x^2], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[(2 - n) x] BesselI[0, 2 Sqrt[1 - n] x], {x, 0, n}], {n, 0, 19}]
Table[Hypergeometric2F1[-n, -n, 1, 1 - n], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)^2*(1-n)^k); \\ Michel Marcus, Jun 01 2020

Formula

a(n) = central coefficient of (1 - (n - 2)*x - (n - 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 + 2*(n - 2)*x + n^2*x^2).
a(n) = n! * [x^n] exp((2 - n)*x) * BesselI(0,2*sqrt(1 - n)*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (1-n)^k.

A333989 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1+(k-1)*x) / (1+2*(k-1)*x+((k+1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -4, 1, 1, -2, -7, 0, 1, 1, -3, -8, 23, 16, 1, 1, -4, -7, 64, 17, 0, 1, 1, -5, -4, 117, -128, -241, -64, 1, 1, -6, 1, 176, -527, -512, 329, 0, 1, 1, -7, 8, 235, -1264, 237, 4096, 1511, 256, 1, 1, -8, 17, 288, -2399, 3776, 11753, -8192, -5983, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2020

Keywords

Examples

			Square array begins:
  1,  1,    1,    1,    1,     1, ...
  1,  0,   -1,   -2,   -3,    -4, ...
  1, -4,   -7,   -8,   -7,    -4, ...
  1,  0,   23,   64,  117,   176, ...
  1, 16,   17, -128, -527, -1264, ...
  1,  0, -241, -512,  237,  3776, ...

Crossrefs

Main diagonal gives A333991.
Cf. A307884, A333988.

Programs

  • Mathematica
    T[n_, 0] := 1; T[n_, k_] := Sum[(-k)^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)
  • PARI
    {T(n, k) = sum(j=0, n, (-k)^j*binomial(2*n, 2*j))}

Formula

T(n,k) = Sum_{j=0..n} (-k)^j * binomial(2*n,2*j).
T(0,k) = 1, T(1,k) = 1-k and T(n,k) = -2 * (k-1) * T(n-1,k) - (k+1)^2 * T(n-2,k) for n>1.
Showing 1-8 of 8 results.

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