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

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

0,4

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A117435 A282252 A268367 * A136448 A166318 A166317

Adjacent sequences: A117433 A117434 A117435 * A117437 A117438 A117439

Keyword

nonn,tabl

Author

Paul Barry, Mar 16 2006

Status

approved