A117436 - OEIS

%I #23 Jun 02 2021 22:17:08

%S 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,

%T 0,27328,0,2800,0,84,0,1,354560,0,109312,0,5600,0,112,0,1,0,3191040,0,

%U 327936,0,10080,0,144,0,1,51733504,0,15955200,0,819840,0,16800,0,180,0,1

%N Triangle related to exp(x)*sec(2*x).

%C Inverse is A117435.

%C 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

%H G. C. Greubel, <a href="/A117436/b117436.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Number triangle whose k-th column has e.g.f. (x^k/k!)*sec(2*x).

%F T(n, 0) = A002436(n).

%F Sum_{k=0..n} T(n, k) = A117437(n).

%F 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

%e Triangle begins as:

%e          1;

%e          0,       1;

%e          4,       0,        1;

%e          0,      12,        0,      1;

%e         80,       0,       24,      0,      1;

%e          0,     400,        0,     40,      0,     1;

%e       3904,       0,     1200,      0,     60,     0,     1;

%e          0,   27328,        0,   2800,      0,    84,     0,   1;

%e     354560,       0,   109312,      0,   5600,     0,   112,   0,   1;

%e          0, 3191040,        0, 327936,      0, 10080,     0, 144,   0, 1;

%e   51733504,       0, 15955200,      0, 819840,     0, 16800,   0, 180, 0, 1;

%t T[n_, k_]:= Binomial[n, k]*(2*I)^(n-k)*EulerE[n-k];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 01 2021 *)

%o (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

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

%K nonn,tabl

%O 0,4

%A _Paul Barry_, Mar 16 2006