OFFSET
0,3
COMMENTS
From Emeric Deutsch, Jun 27 2009: (Start)
T(n,0) = A000364(n), the Euler (or secant) numbers.
Sum of entries in row n = A000281(n).
(End)
LINKS
Alois P. Heinz, Rows n = 0..100, flattened
C. Radoux, The Hankel Determinant of Exponential Polynomials: A Very Short Proof and a New Result Concerning Euler Numbers, Amer. Math. Monthly, 109 (2002), 277-278.
EXAMPLE
Triangle begins:
1;
1, 2;
5, 28, 24;
61, 662, 1320, 720;
1385, 24568, 83664, 100800, 40320;
50521, 1326122, 6749040, 13335840, 11491200, 3628800;
MAPLE
A151775 := proc(n, k) if n= 0 then 1 ; else taylor( (tan(x))^(2*k)/cos(x), x=0, 2*n+1) ; diff(%, x$(2*n)) ; coeftayl(%, x=0, 0) ; fi; end: for n from 0 to 10 do for k from 0 to n do printf("%d ", A151775(n, k)) ; od: printf("\n") ; od: # R. J. Mathar, Jun 24 2009
T := proc (n, k) if n = 0 and k = 0 then 1 elif n = 0 then 0 else simplify(subs(x = 0, diff(tan(x)^(2*k)/cos(x), `$`(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
# alternative Maple program:
T:= (n, k)-> (2*n)!*coeff(series(tan(x)^(2*k)/cos(x), x, 2*n+1), x, 2*n):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 06 2017
MATHEMATICA
T[n_, k_] := (2n)! SeriesCoefficient[Tan[x]^(2k)/Cos[x], {x, 0, 2n}];
T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jun 24 2009, at the suggestion of Alexander R. Povolotsky
EXTENSIONS
More values from R. J. Mathar and Emeric Deutsch, Jun 24 2009
STATUS
approved