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.

A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

Original entry on oeis.org

1, -3, -1, 5, 7, -3, -17, -11, 23, 45, -1, -91, -89, 93, 271, 85, -457, -627, 287, 1541, 967, -2115, -4049, 181, 8279, 7917, -8641, -24475, -7193, 41757, 56143, -27371, -139657, -84915, 194399, 364229, -24569, -753027, -703889, 802165, 2209943
Offset: 0

Views

Author

Paul Barry, Nov 03 2009

Keywords

Comments

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.
Variants are A107920 and A001607.

Links

Programs

  • Mathematica
    a[n_] := Sin[n*ArcTan[Sqrt[7]]]; FullSimplify[Join[{1}, Table[- (2^(n/2 + 1)/Sqrt[7])*(2*a[n] + Sqrt[2]*a[n + 1]), {n, 1, 100}]]] (* or *) Join[{1}, LinearRecurrence[{1,-2},{-3,-1},100]] (* G. C. Greubel, Jun 13 2016 *)

Formula

G.f.: (1-4x+4x^2)/(1-x+2x^2).
From G. C. Greubel, Jun 13 2016: (Start)
a(n) = a(n-1) - 2*a(n-2).
a(n) = -(2^((n+2)/2)/sqrt(7))*( 2*sin(n*arctan(sqrt(7))) + sqrt(2)*sin((n+1)*arctan(sqrt(7))) ), n>=1, and a(0)=1. (End)

A167434 Diagonal sums of the Riordan array (1-4*x+4*x^2, x*(1-2*x)) (A167431).

Original entry on oeis.org

1, -4, 5, -6, 13, -16, 25, -42, 57, -92, 141, -206, 325, -488, 737, -1138, 1713, -2612, 3989, -6038, 9213, -14016, 21289, -32442, 49321, -75020, 114205, -173662, 264245, -402072, 611569, -930562, 1415713, -2153700, 3276837, -4985126, 7584237
Offset: 0

Views

Author

Paul Barry, Nov 03 2009

Keywords

Links

Crossrefs

Cf. A052947, A159287.

Programs

  • Magma
    I:=[1,-4,5]; [n le 3 select I[n] else Self(n-2) - 2*Self(n-3): n in [1..40]]; // G. C. Greubel, Jun 27 2018
  • Mathematica
    LinearRecurrence[{0, 1, -2}, {1, -4, 5}, 100] (* G. C. Greubel, Jun 13 2016 *)
CoefficientList[Series[(1-4x+4x^2)/(1-x^2+2x^3),{x,0,40}],x] (* Harvey P. Dale, Nov 08 2022 *)
  • PARI
    x='x+O('x^40); Vec((1-4*x+4*x^2)/(1-x^2+2*x^3)) \\ G. C. Greubel, Jun 27 2018

Formula

G.f.: (1-2*x)^2/(1-x^2+2*x^3).
a(n) = (-1)^n*A052947(n+4). - R. J. Mathar, Jun 24 2024

A167432 Riordan array (c(2x)^2,xc(2x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 4, 1, 20, 6, 1, 112, 36, 8, 1, 672, 224, 56, 10, 1, 4224, 1440, 384, 80, 12, 1, 27456, 9504, 2640, 600, 108, 14, 1, 183040, 64064, 18304, 4400, 880, 140, 16, 1, 1244672, 439296, 128128, 32032, 6864, 1232, 176, 18, 1, 8599552, 3055104, 905216, 232960
Offset: 0

Views

Author

Paul Barry, Nov 03 2009

Keywords

Comments

Inverse of (1-4x+4x^2,x(1-2x)) (A167431). Row sums are A084076. First column is A003645.

Examples

			Triangle begins
1,
4, 1,
20, 6, 1,
112, 36, 8, 1,
672, 224, 56, 10, 1,
4224, 1440, 384, 80, 12, 1,
27456, 9504, 2640, 600, 108, 14, 1,
183040, 64064, 18304, 4400, 880, 140, 16, 1,
1244672, 439296, 128128, 32032, 6864, 1232, 176, 18, 1,
8599552, 3055104, 905216, 232960, 52416, 10192, 1664, 216, 20, 1,
60196864, 21498880, 6449664, 1697280, 396032, 81536, 14560, 2184, 260, 22, 1
The production matrix is
4, 1,
4, 2, 1,
8, 4, 2, 1,
16, 8, 4, 2, 1,
32, 16, 8, 4, 2, 1,
64, 32, 16, 8, 4, 2, 1,
128, 64, 32, 16, 8, 4, 2, 1,
256, 128, 64, 32, 16, 8, 4, 2, 1,
512, 256, 128, 64, 32, 16, 8, 4, 2, 1
When topped with the row (1,0,0,0...), this has inverse
1,
-4, 1,
4, -2, 1,
0, 0, -2, 1,
0, 0, 0, -2, 1,
0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, 0, 0, -2, 1

Formula

Number triangle T(n,k)=A054445(n,k)*2^(n-k).

A329918 Coefficients of orthogonal polynomials related to the Jacobsthal numbers A152046, triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 4, 0, 6, 0, 1, 0, 0, 12, 0, 8, 0, 1, 0, 8, 0, 24, 0, 10, 0, 1, 0, 0, 32, 0, 40, 0, 12, 0, 1, 0, 16, 0, 80, 0, 60, 0, 14, 0, 1, 0, 0, 80, 0, 160, 0, 84, 0, 16, 0, 1, 0, 32, 0, 240, 0, 280, 0, 112, 0, 18, 0, 1
Offset: 0

Views

Author

Peter Luschny, Nov 28 2019

Keywords

Examples

			Triangle starts:
  [0] 1;
  [1] 0,  1;
  [2] 0,  0,  1;
  [3] 0,  2,  0,  1;
  [4] 0,  0,  4,  0,  1;
  [5] 0,  4,  0,  6,  0,  1;
  [6] 0,  0, 12,  0,  8,  0,  1;
  [7] 0,  8,  0, 24,  0, 10,  0,  1;
  [8] 0,  0, 32,  0, 40,  0, 12,  0, 1;
  [9] 0, 16,  0, 80,  0, 60,  0, 14, 0, 1;
The first few polynomials:
  p(0,x) = 1;
  p(1,x) = x;
  p(2,x) = x^2;
  p(3,x) = 2*x + x^3;
  p(4,x) = 4*x^2 + x^4;
  p(5,x) = 4*x + 6*x^3 + x^5;
  p(6,x) = 12*x^2 + 8*x^4 + x^6;

Crossrefs

Row sums are A001045 starting with 1, which is A152046. These are in signed form also the alternating row sums. Diagonal sums are aerated A133494.
Cf. A110509, A113953, A114192, A167431, A322942.

Programs

  • Julia
    using Nemo # Returns row n.
function A329918(row)
    R, x = PolynomialRing(ZZ, "x")
    function p(n)
        n < 3 && return x^n
        x*p(n-1) + 2*p(n-2)
    end
    p = p(row)
    [coeff(p, k) for k in 0:row]
end
for row in 0:9 println(A329918(row)) end # prints triangle
  • Maple
    T := (n, k) -> `if`((n+k)::odd, 0, 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..11);

Formula

p(n) = x*p(n-1) + 2*p(n-2) for n >= 3; p(0) = 1, p(1) = x, p(2) = x^2.
T(n, k) = [x^k] p(n).
T(n, k) = 2^((n-k)/2)*binomial((n+k)/2-1, (n-k)/2) if n+k is even otherwise 0.
Showing 1-4 of 4 results.

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