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
Examples
Triangle begins:
1;
1, 2;
5, 28, 24;
61, 662, 1320, 720;
1385, 24568, 83664, 100800, 40320;
50521, 1326122, 6749040, 13335840, 11491200, 3628800;
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.
Crossrefs
A subtriangle of A008294.
Programs
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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
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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 *)
Extensions
More values from R. J. Mathar and Emeric Deutsch, Jun 24 2009
Comments