A108124 - OEIS

A108124

E.g.f. x/(1+sin(x)).

0, 1, -2, 6, -20, 80, -366, 1904, -11080, 71424, -505210, 3891712, -32433180, 290787328, -2791053734, 28556359680, -310264194320, 3567710830592, -43287834157938, 552688817143808, -7407423764750500, 103981459115606016, -1525675236649033822, 23354749389657604096

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

FORMULA

E.g.f. x/(1+sin(x)).

From Sergei N. Gladkovskii, May 29 2012: (Start)

Let E(x) be the e.g.f., then

E(x) = 1 - 1/(1+x)+ x^4/((1+x)*((1+x)*G(0) + x^2)) where G(k)= (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction Euler's kind, 1-step ).

E(x) = 1 - 1/(1+x)+ x^4/((1+x)*((1+x)*G(0) + x^2)) where G(k)= 8*k+6-x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction Euler's 2 kind, 2-step ).

E(x) = x/(1+x*G(0)) where G(k)= 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction 3 kind, 3-step ).

(End)

E.g.f. x/(1 + x*G(0)) where G(k)= 1 + x^2/( (2*k+1)*(2*k+3) - 2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 01 2012

MATHEMATICA

g[x_] = FullSimplify[x/(-1 + Sum[(-1)^n*x^(2*n - 1)/(2*n - 1)!, {n, 1, Infinity}])] h[x_, n_] = Dt[g[x], {x, n}] b[x_] = Table[h[x, n], {n, 0, 50}]; b[0]

With[{nn=30}, CoefficientList[Series[x/(1+Sin[x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 14 2014 *)

PROG

(Sage)

@CachedFunction

def c(n, k) :

if n==k: return 1

if k<1 or k>n: return 0

return ((n-k)//2+1)*c(n-1, k-1)+k*c(n-1, k+1)

def A108124(n): return (-1)^(n+1)*n*add(c(n, k) for k in (0..n))

[A108124(n) for n in (0..23)] # Peter Luschny, Jun 10 2014

CROSSREFS

Cf. A186365