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From the expansion of sin(sin(x)*x).
1

%I #26 Apr 01 2017 14:31:35

%S 0,2,-4,-114,3352,-35270,-2244012,198654470,-8021832016,-150983244558,

%T 67525484385580,-7526828271926018,368068475511786696,

%U 48206694242241834026,-16586068178557581107068,2563355081796258270543990,-153878422314204916436611232

%N From the expansion of sin(sin(x)*x).

%F sin(x sin x) = Sum a(n) x^(2n)/(2n)!. - _N. J. A. Sloane_, Aug 28 2012

%F a(n) = sum(k=0..n, binomial(2*n,2*k+1)*(4^(-k)*sum(i=0..k, (2*i-2*k-1)^(2*n-2*k-1)*binomial(2*k+1,i)*(-1)^(n-i+k)))). - _Vladimir Kruchinin_, Jun 28 2011

%e sin(sin(x)*x) = x^2-(1/6)*x^4-(19/120)*x^6+(419/5040)*x^8-(3527/362880)*x^10-(187001/39916800)*x^12+... - _N. J. A. Sloane_, Aug 28 2012

%t With[{nn=40},Take[CoefficientList[Series[Sin[Sin[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Aug 28 2012 *)

%o (Maxima)

%o a(n):=sum(binomial(2*n,2*k+1)*(4^(-k)*sum((2*i-2*k-1)^(2*n-2*k-1)*binomial(2*k+1,i)*(-1)^(n-i+k),i,0,k)),k,0,n); /* _Vladimir Kruchinin_, Jun 28 2011 */

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997