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

1, 1, 2, 5, 28, 24, 61, 662, 1320, 720, 1385, 24568, 83664, 100800, 40320, 50521, 1326122, 6749040, 13335840, 11491200, 3628800, 2702765, 98329108, 692699304, 1979524800, 2739623040, 1836172800, 479001600, 199360981, 9596075582

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

A subtriangle of A008294.

Cf. A000364, A000281. [Emeric Deutsch, Jun 27 2009]

Sequence in context: A208227 A127357 A025170 * A286879 A326230 A095159

Adjacent sequences: A151772 A151773 A151774 * A151776 A151777 A151778

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