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 1-3 of 3 results.

A117436 Triangle related to exp(x)*sec(2*x).

Original entry on oeis.org

1, 0, 1, 4, 0, 1, 0, 12, 0, 1, 80, 0, 24, 0, 1, 0, 400, 0, 40, 0, 1, 3904, 0, 1200, 0, 60, 0, 1, 0, 27328, 0, 2800, 0, 84, 0, 1, 354560, 0, 109312, 0, 5600, 0, 112, 0, 1, 0, 3191040, 0, 327936, 0, 10080, 0, 144, 0, 1, 51733504, 0, 15955200, 0, 819840, 0, 16800, 0, 180, 0, 1
Offset: 0

Views

Author

Paul Barry, Mar 16 2006

Keywords

Comments

Inverse is A117435.
Conjecture: The d-th diagonal (starting with d=0) is proportional to the sequence of generalized binomial coefficients binomial(-x, d) where x is the column index. - Søren G. Have, Feb 26 2017

Examples

			Triangle begins as:
         1;
         0,       1;
         4,       0,        1;
         0,      12,        0,      1;
        80,       0,       24,      0,      1;
         0,     400,        0,     40,      0,     1;
      3904,       0,     1200,      0,     60,     0,     1;
         0,   27328,        0,   2800,      0,    84,     0,   1;
    354560,       0,   109312,      0,   5600,     0,   112,   0,   1;
         0, 3191040,        0, 327936,      0, 10080,     0, 144,   0, 1;
  51733504,       0, 15955200,      0, 819840,     0, 16800,   0, 180, 0, 1;

Links

Crossrefs

Cf. A000364, A002436 (1st column), A117435 (inverse), A117437 (row sums).

Programs

  • Mathematica
    T[n_, k_]:= Binomial[n, k]*(2*I)^(n-k)*EulerE[n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)
  • Sage
    flatten([[binomial(n,k)*(2*i)^(n-k)*euler_number(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Formula

Number triangle whose k-th column has e.g.f. (x^k/k!)*sec(2*x).
T(n, 0) = A002436(n).
Sum_{k=0..n} T(n, k) = A117437(n).
T(n, k) = binomial(n,k) * (2*i)^(n-k) * E(n-k), where E(n) are the Euler numbers with E(2*n) = A000364(n) and E(2*n+1) = 0. - G. C. Greubel, Jun 01 2021

A117438 Triangle T(n, k) = binomial(2*n-k, k)*(-4)^(n-k), read by rows.

Original entry on oeis.org

1, -4, 1, 16, -12, 1, -64, 80, -24, 1, 256, -448, 240, -40, 1, -1024, 2304, -1792, 560, -60, 1, 4096, -11264, 11520, -5376, 1120, -84, 1, -16384, 53248, -67584, 42240, -13440, 2016, -112, 1, 65536, -245760, 372736, -292864, 126720, -29568, 3360, -144, 1
Offset: 0

Views

Author

Paul Barry, Mar 16 2006

Keywords

Examples

			Triangle begins
      1;
     -4,      1;
     16,    -12,     1;
    -64,     80,   -24,     1;
    256,   -448,   240,   -40,    1;
  -1024,   2304, -1792,   560,  -60,   1;
   4096, -11264, 11520, -5376, 1120, -84, 1;

Links

Crossrefs

Cf. A117435, A117439.

Programs

  • Mathematica
    Table[Binomial[2*n-k, k]*(-4)^(n-k), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)
  • Sage
    flatten([[binomial(2*n-k, k)*(-4)^(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Formula

T(n, k) = binomial(2*n-k, k)*(-4)^(n-k).
Sum_{k=0..n} T(n, k) = (-1)^n*(2*n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A117439(n) (upward diagonal sums).
T(n, k) = A117435(2*n-k, k).

A166317 Exponential Riordan array [sec(2x), arctanh(tan(x))].

Original entry on oeis.org

1, 0, 1, 4, 0, 1, 0, 16, 0, 1, 80, 0, 40, 0, 1, 0, 640, 0, 80, 0, 1, 3904, 0, 2800, 0, 140, 0, 1, 0, 49152, 0, 8960, 0, 224, 0, 1, 354560, 0, 319744, 0, 23520, 0, 336, 0, 1, 0, 6225920, 0, 1454080, 0, 53760, 0, 480, 0, 1, 51733504, 0, 54897920, 0, 5230720, 0, 110880, 0, 660, 0, 1
Offset: 0

Views

Author

Paul Barry, Oct 11 2009

Keywords

Comments

The Bell transform of abs(2^n*euler_number(n)). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			Triangle begins
  1;
  0, 1;
  4, 0, 1;
  0, 16, 0, 1;
  80, 0, 40, 0, 1;
  0, 640, 0, 80, 0, 1;
  3904, 0, 2800, 0, 140, 0, 1;
  0, 49152, 0, 8960, 0, 224, 0, 1;
  354560, 0, 319744, 0, 23520, 0, 336, 0, 1;
  0, 6225920, 0, 1454080, 0, 53760, 0, 480, 0, 1;
  51733504, 0, 54897920, 0, 5230720, 0, 110880, 0, 660, 0, 1;
Production matrix is
    0,    1;
    4,    0,    1;
    0,   12,    0,    1;
   16,    0,   24,    0,    1;
    0,   80,    0,   40,    0,    1;
   64,    0,  240,    0,   60,    0,   1;
    0,  448,    0,  560,    0,   84,   0,   1;
  256,    0, 1792,    0, 1120,    0, 112,   0, 1;
    0, 2304,    0, 5376,    0, 2016,   0, 144, 0, 1;
which is the exponential Riordan array [cosh(2x),x] minus its top row. (Cf. also A117435.)

Crossrefs

Row sums are A012259(n+1).
Inverse is A166318 which is a signed version of this sequence.
Cf. A117435, A264428.

Programs

  • Mathematica
    (* The function BellMatrix is defined in A264428. *)
rows = 12;
M = BellMatrix[Abs[2^#*EulerE[#]]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jul 11 2019 *)
  • Sage
    # uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: abs(2^n*euler_number(n)), 10) # Peter Luschny, Jan 18 2016
Showing 1-3 of 3 results.

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