%I M5023 N2166 #30 Sep 15 2024 14:41:05
%S 16,272,3968,56320,814080,12207360,191431680,3149752320,54428774400,
%T 987559372800,18797300121600,374883257548800,7822865085235200,
%U 170560590520320000,3879770715684864000,91945674412720128000
%N Generalized tangent numbers.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Letterio Toscano, Sulla Derivata di Ordinen della Funzione tg(x), Tohoku Math. J., 42 (1936), 144-154.
%F Ignoring the initial term a(4) = 16 and working with an offset of 0 the e.g.f. appears to be the rational function 16*(17+78*t+45*t^2)/(1-t)^10 = 272 + 3968*t + 56320*t^2/2! + ... . - _Peter Bala_, Apr 23 2012
%F This rational function occurs in the series reversion (x-t*tan(x))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3! + 8*t*(2+3*t)/(1-t)^7*x^5/5! + 16*t*(17+78*t+45*t^2)/(1-t)^10*x^7/7! + ..., which is the e.g.f. for the triangle A059419 read by diagonals. - _Peter Bala_, Apr 23 2012
%o (PARI) a(n, k)=if(k<0,0,if(n==1 && k==1,1,if(k>n,0,(k-1)*a(n-1,k-1)+(k+1)*a(n-1,k+1))))
%o for(n=0,25,print1(a(n+6, n)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006
%Y Cf. A059419.
%K nonn
%O 4,1
%A _N. J. A. Sloane_
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006