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%I #10 Aug 10 2019 02:49:39
%S 1,-1,1,5,-10,5,-61,183,-183,61,1385,-5540,8310,-5540,1385,-50521,
%T 252605,-505210,505210,-252605,50521,2702765,-16216590,40541475,
%U -54055300,40541475,-16216590,2702765,-199360981,1395526867,-4186580601,6977634335,-6977634335,4186580601,-1395526867,199360981
%N Triangle with Euler (secant) numbers, read by rows, T(n, k) for 0 <= k <= n.
%F T(n, k) = (2*n)! [x^k] [y^(2*n)] sec(y*sqrt(x - 1)).
%F Sum_{k=0..n} (-1)^(n-k)*T(n, k) = |A012816(n+1)|.
%e Triangle starts:
%e [0] 1;
%e [1] -1, 1;
%e [2] 5, -10, 5;
%e [3] -61, 183, -183, 61;
%e [4] 1385, -5540, 8310, -5540, 1385;
%e [5] -50521, 252605, -505210, 505210, -252605, 50521;
%e [6] 2702765, -16216590, 40541475, -54055300, 40541475, -16216590, 2702765;
%t gf := Sec[y Sqrt[x - 1]]; ser := Series[gf, {y, 0, 26}];
%t cy[n_] := n! Coefficient[ser, y, n]; row[n_] := CoefficientList[cy[2 n], x];
%t Table[row[n], {n, 0, 7}] // Flatten
%Y Cf. A000364, A012816, A326722.
%K sign,tabl
%O 0,4
%A _Peter Luschny_, Aug 06 2019