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.

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -3, -5, 1, 1, 1, -5, -11, -5, 1, 1, 1, -7, -17, 1, 11, 1, 1, 1, -9, -23, 19, 81, 41, 1, 1, 1, -11, -29, 49, 211, 141, 29, 1, 1, 1, -13, -35, 91, 401, 181, -363, -125, 1, 1, 1, -15, -41, 145, 651, 41, -2015, -1791, -365, 1
Offset: 0

Views

Author

Seiichi Manyama, May 02 2019

Keywords

Examples

			Square array begins:
   1,  1,    1,     1,     1,      1,      1, ...
   1,  1,    1,     1,     1,      1,      1, ...
   1, -1,   -3,    -5,    -7,     -9,    -11, ...
   1, -5,  -11,   -17,   -23,    -29,    -35, ...
   1, -5,    1,    19,    49,     91,    145, ...
   1, 11,   81,   211,   401,    651,    961, ...
   1, 41,  141,   181,    41,   -399,  -1259, ...
   1, 29, -363, -2015, -5767, -12459, -22931, ...

Links

Crossrefs

Columns k=0..5 give A000012, A098331, A098332, A098333, A098334.
Main diagonal gives A307862.
Cf. A307819, A307855.

Programs

  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

Formula

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

A116093 Expansion of 1/sqrt(1+4*x+12*x^2).

Original entry on oeis.org

1, -2, 0, 16, -56, 48, 384, -1920, 3168, 8512, -66560, 161280, 113920, -2224640, 7311360, -3354624, -69253632, 306754560, -408059904, -1898029056, 12054196224, -25377005568, -38874316800, 443400781824, -1289598418944, -52751204352, 15086928789504, -58620595404800
Offset: 0

Views

Author

Paul Barry, Feb 04 2006

Keywords

Comments

Apart from signs identical this is to A098336. - Joerg Arndt, Jun 30 2013
Fourth binomial transform of the expansion of 1/sqrt(1-4*x+12*x^2), A098336.

Links

Crossrefs

Column 2 of A307819.
Cf. A098336.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+12*x^2) )); // G. C. Greubel, May 10 2019
  • Mathematica
    CoefficientList[Series[1/Sqrt[1+4x+12x^2],{x,0,30}],x] (* Harvey P. Dale, Oct 15 2014 *)
  • PARI
    my(x='x+O('x^30)); Vec(1/sqrt(1+4*x+12*x^2)) \\ G. C. Greubel, May 10 2019
  • Sage
    (1/sqrt(1+4*x+12*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019

Formula

E.g.f.: exp(-2*x)*Bessel_I(0,2*sqrt(-2)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)(-2)^(n-k).
D-finite with recurrence: n*a(n) +2*(2*n-1)*a(n-1) +12*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 07 2012
G.f.: G(0), where G(k)= 1 - 2*x*(1+3*x)*(4*k+1)/( 2*k+1 - x*(1+3*x)*(2*k+1)*(4*k+3)/(x*(1+3*x)*(4*k+3) - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 30 2013

A307910 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, 3, 0, 1, 3, 8, 7, 0, 1, 4, 15, 32, 19, 0, 1, 5, 24, 81, 136, 51, 0, 1, 6, 35, 160, 459, 592, 141, 0, 1, 7, 48, 275, 1120, 2673, 2624, 393, 0, 1, 8, 63, 432, 2275, 8064, 15849, 11776, 1107, 0, 1, 9, 80, 637, 4104, 19375, 59136, 95175, 53344, 3139, 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,   3,     8,    15,     24,      35,      48, ...
   0,   7,    32,    81,    160,     275,     432, ...
   0,  19,   136,   459,   1120,    2275,    4104, ...
   0,  51,   592,  2673,   8064,   19375,   40176, ...
   0, 141,  2624, 15849,  59136,  168125,  400896, ...
   0, 393, 11776, 95175, 439296, 1478125, 4053888, ...

Links

Crossrefs

Columns k=0..4 give A000007, A002426, A006139, A122868, A059304.
Main diagonal gives A092366.
Cf. A107267, A292627, A307819, A307847, A307855, A307883.

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).

A307911 Coefficient of x^n in expansion of (1 - n*x - n*x^2)^n.

Original entry on oeis.org

1, -1, 0, 27, -416, 5625, -74304, 924385, -8626176, -48361131, 7124800000, -340421390199, 13686496542720, -522760216822129, 19658830846298112, -735037915447265625, 27218267709730979840, -980444996625142158435, 32830565919734078521344, -889052809376495994642527
Offset: 0

Views

Author

Seiichi Manyama, May 05 2019

Keywords

Comments

Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*n*x + n*(n+4)*x^2).

Links

Crossrefs

Main diagonal of A307819.
Cf. A092366, A307862.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * Binomial[2*k, k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Join[{1}, Table[(-n)^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -4/n], {n, 1, 20}]] (* Vaclav Kotesovec, May 12 2021 *)
  • PARI
    {a(n) = polcoef((1-n*x-n*x^2)^n, n)}
  • PARI
    {a(n) = sum(k=0, n\2, (-n)^(n-k)*binomial(n, k)*binomial(n-k, k))}
  • PARI
    {a(n) = sum(k=0, n\2, (-n)^(n-k)*binomial(n, 2*k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,k) * binomial(n-k,k) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,2*k) * binomial(2*k,k).
For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 1, -4/n). - Vaclav Kotesovec, May 12 2021

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 1-5 of 5 results.

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