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A151775 Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.

%I A151775 #15 Dec 10 2019 12:10:39
%S A151775 1,1,2,5,28,24,61,662,1320,720,1385,24568,83664,100800,40320,50521,
%T A151775 1326122,6749040,13335840,11491200,3628800,2702765,98329108,692699304,
%U A151775 1979524800,2739623040,1836172800,479001600,199360981,9596075582
%N A151775 Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.
%C A151775 From _Emeric Deutsch_, Jun 27 2009: (Start)
%C A151775 T(n,0) = A000364(n), the Euler (or secant) numbers.
%C A151775 Sum of entries in row n = A000281(n).
%C A151775 (End)
%H A151775 Alois P. Heinz, <a href="/A151775/b151775.txt">Rows n = 0..100, flattened</a>
%H A151775 C. Radoux, <a href="http://www.jstor.org/stable/2695357">The Hankel Determinant of Exponential Polynomials: A Very Short Proof and a New Result Concerning Euler Numbers</a>, Amer. Math. Monthly, 109 (2002), 277-278.
%e A151775 Triangle begins:
%e A151775       1;
%e A151775       1,       2;
%e A151775       5,      28,      24;
%e A151775      61,     662,    1320,      720;
%e A151775    1385,   24568,   83664,   100800,    40320;
%e A151775   50521, 1326122, 6749040, 13335840, 11491200, 3628800;
%p A151775 A151775 := proc(n,k) if n= 0 then 1 ; else taylor( (tan(x))^(2*k)/cos(x),x=0,2*n+1) ; diff(%,x$(2*n)) ; coeftayl(%,x=0,0) ; fi; end: for n from 0 to 10 do for k from 0 to n do printf("%d ", A151775(n,k)) ; od: printf("\n") ; od: # _R. J. Mathar_, Jun 24 2009
%p A151775 T := proc (n, k) if n = 0 and k = 0 then 1 elif n = 0 then 0 else simplify(subs(x = 0, diff(tan(x)^(2*k)/cos(x), `$`(x, 2*n)))) end if end proc: for n from 0 to 7 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form; _Emeric Deutsch_, Jun 27 2009
%p A151775 # alternative Maple program:
%p A151775 T:= (n, k)-> (2*n)!*coeff(series(tan(x)^(2*k)/cos(x), x, 2*n+1), x, 2*n):
%p A151775 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Aug 06 2017
%t A151775 T[n_, k_] := (2n)! SeriesCoefficient[Tan[x]^(2k)/Cos[x], {x, 0, 2n}];
%t A151775 T[0, 0] = 1;
%t A151775 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 10 2019, after _Alois P. Heinz_ *)
%Y A151775 A subtriangle of A008294.
%Y A151775 Cf. A000364, A000281. [_Emeric Deutsch_, Jun 27 2009]
%K A151775 nonn,tabl
%O A151775 0,3
%A A151775 _N. J. A. Sloane_, Jun 24 2009, at the suggestion of _Alexander R. Povolotsky_
%E A151775 More values from _R. J. Mathar_ and _Emeric Deutsch_, Jun 24 2009