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.

A128417 Number triangle T(n,k) = 2^(n-k)*C(2*n,n-k).

Original entry on oeis.org

1, 4, 1, 24, 8, 1, 160, 60, 12, 1, 1120, 448, 112, 16, 1, 8064, 3360, 960, 180, 20, 1, 59136, 25344, 7920, 1760, 264, 24, 1, 439296, 192192, 64064, 16016, 2912, 364, 28, 1, 3294720, 1464320, 512512, 139776, 29120, 4480, 480, 32, 1
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Inverse of A128414. Row sums are A128418. Diagonal sums are A128419.

Examples

			Triangle begins:
  1,
  4, 1,
  24, 8, 1,
  160, 60, 12, 1,
  1120, 448, 112, 16, 1,
  8064, 3360, 960, 180, 20, 1,
  59136, 25344, 7920, 1760, 264, 24, 1,
  439296, 192192, 64064, 16016, 2912, 364, 28, 1
  ...

Links

Crossrefs

Cf. A128413.

Programs

  • Mathematica
    Flatten[Table[2^(n-k) Binomial[2n,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Nov 02 2011 *)

Formula

Riordan array (1/sqrt(1-8*x),(1-4*x-sqrt(1-8*x))/(8*x)).
T(n,k) = 2^(n-k)*A094527(n,k).

A128415 Expansion of (1-4x^2)/(1+3x+4x^2).

Original entry on oeis.org

1, -3, 1, 9, -31, 57, -47, -87, 449, -999, 1201, 393, -5983, 16377, -25199, 10089, 70529, -251943, 473713, -413367, -654751, 3617721, -8234159, 10231593, 2241857, -47651943, 133988401, -211357431, 98118689
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Row sums of number triangle A128414.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1 - 4 x^2) / (1 + 3 x + 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
LinearRecurrence[{-3,-4},{1,-3,1},40] (* Harvey P. Dale, Sep 24 2022 *)

Formula

For n>0, a(n) = (1/r)^n + (1/s)^n, with r = (-3-i*sqrt(7))/8 and s = (-3+i*sqrt(7))/8 the roots of 4x^2+3x+1. - Ralf Stephan, Jul 20 2013
a(n) = -3*a(n-1) - 4*a(n-2) for n > 2. - Harry Richman, May 05 2020

A128416 Expansion of (1-4x^2)/(1+4x+3x^2).

Original entry on oeis.org

1, -4, 9, -24, 69, -204, 609, -1824, 5469, -16404, 49209, -147624, 442869, -1328604, 3985809, -11957424, 35872269, -107616804, 322850409, -968551224, 2905653669, -8716961004, 26150883009, -78452649024, 235357947069, -706073841204, 2118221523609
Offset: 0

Views

Author

Paul Barry, Mar 02 2007

Keywords

Comments

Diagonal sums of number triangle A128414.

Links

Crossrefs

Cf. A128414.

Programs

  • PARI
    Vec((1-4*x^2)/(1+4*x+3*x^2) + O(x^30)) \\ Colin Barker, Sep 21 2016

Formula

G.f.: (1-4*x^2)/(1+4*x+3*x^2).
For n > 1, |a(n)| = 3*(|a(n-1)| - 1). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Sep 17 2010
From Colin Barker, Sep 21 2016: (Start)
a(n) = (-1)^n*(9+5*3^n)/6 for n>0.
a(n) = -4*a(n-1)-3*a(n-2) for n>2. (End)

A137374 Triangular array of the coefficients of the Jacobsthal-Lucas polynomials ordered by descending powers, read by rows.

Original entry on oeis.org

2, 1, 4, 1, 6, 1, 8, 8, 1, 20, 10, 1, 16, 36, 12, 1, 56, 56, 14, 1, 32, 128, 80, 16, 1, 144, 240, 108, 18, 1, 64, 400, 400, 140, 20, 1, 352, 880, 616, 176, 22, 1, 128, 1152, 1680, 896, 216, 24, 1, 832, 2912, 2912, 1248, 260, 26, 1, 256, 3136, 6272, 4704, 1680, 308, 28, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 09 2008

Keywords

Comments

The even rows which start with 4, 8, 16 ... appear to be the absolute values of the Riordan array A128414. - Georg Fischer, Feb 25 2020

Examples

			The triangle starts:
    2;
    1;
    4,   1;
    6,   1;
    8,   8,   1;
   20,  10,   1;
   16,  36,  12,   1;
   56,  56,  14,   1;
   32, 128,  80,  16,  1;
  144, 240, 108,  18,  1;
   64, 400, 400, 140, 20, 1;
  352, 880, 616, 176, 22, 1;
  ...

Links

Crossrefs

Row sums give A014551.
Cf. A034807.

Programs

  • Maple
    b:= proc(n) option remember;
      `if`(n<2, 2-n, b(n-1)+2*expand(x*b(n-2)))
    end:
T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Feb 25 2020
  • Mathematica
    f[0] = 2; f[1] = 1; f[n_] := 2 x f[n - 2] + f[n - 1];
Table[Reverse[CoefficientList[f[n], x]], {n, 0, 14}] // Flatten (* Jinyuan Wang, Feb 25 2020 *)

Formula

Let p(n, x) = 2*x*p(n-2, x) + p(n-1, x) with p(0, x) = 2 and p(1, x) = 1. The coefficients of the polynomial p(n, x), listed in reverse order, give row n. - Jinyuan Wang, Feb 25 2020

Extensions

Offset set to 0 by Peter Luschny, Feb 25 2020
Showing 1-4 of 4 results.

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