A002303 Generalized tangent numbers.

16, 272, 3968, 56320, 814080, 12207360, 191431680, 3149752320, 54428774400, 987559372800, 18797300121600, 374883257548800, 7822865085235200, 170560590520320000, 3879770715684864000, 91945674412720128000

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Letterio Toscano, Sulla Derivata di Ordinen della Funzione tg(x), Tohoku Math. J., 42 (1936), 144-154.

Crossrefs

Cf. A059419.

Programs

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))))
for(n=0,25,print1(a(n+6, n)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006

Formula

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

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

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006