A008294 Triangle of coefficients in expansion of D^n (sec x) / sec x in powers of tan x.
1, 1, 1, 2, 5, 6, 5, 28, 24, 61, 180, 120, 61, 662, 1320, 720, 1385, 7266, 10920, 5040, 1385, 24568, 83664, 100800, 40320, 50521, 408360, 1023120, 1028160, 362880, 50521, 1326122, 6749040, 13335840, 11491200, 3628800, 2702765, 30974526, 113760240
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013-2015.
- Dominique Foata and Guo-Niu Han, Multivariable Tangent and Secant q-derivative Polynomials. - _N. J. A. Sloane_, Oct 05 2012
- J. Francon, Histoires de fichiers, RAIRO Informatique Théorique et Applications, 12 (1978), 49-62.
- J. Francon, Histoires de fichiers, RAIRO Informatique Théorique et Applications, 12 (1978), 49-62. (Annotated scanned copy)
- Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180.
- Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
Programs
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Mathematica
nmax = 11; t[0, 0] = 1; t[0, k_] = 0; t[n_, k_] := t[n, k] = k*t[n-1, k-1] + (k+1)*t[n-1, k+1]; Flatten[ Table[ t[n, k-1], {n, 0, nmax}, {k, Mod[n, 2]+1, n+1, 2}]] (* Jean-François Alcover, Nov 08 2011 *)
Formula
a(0, k) = delta(0, k); a(n+1, k) = k*a(n, k-1) + (k+1)*a(n, k+1).