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A218798 E.g.f.: Sum_{n>=0} a(n) * (cos(n*x)-sin(n*x)) * x^n/n! = 1 + x.

%I A218798 #20 Jun 01 2017 04:39:48
%S A218798 1,1,2,15,224,5665,215136,11466903,815542272,74626924833,
%T A218798 8539305740800,1194616544819423,200588161750990848,
%U A218798 39810821495648979009,9219256372209251966976,2463653490073311266825895,752468103154551980520636416,260483894045203867466646001345
%N A218798 E.g.f.: Sum_{n>=0} a(n) * (cos(n*x)-sin(n*x)) * x^n/n!  =  1 + x.
%C A218798 Compare to the LambertW identity:
%C A218798 1 + Sum_{n>=1} n^(n-1) * exp(-n*x) * x^n/n!  =  1 + x.
%C A218798 Limit A219504(n)/A218798(n) = 2.30118311046652539351786883792086321360311554689487793288...
%H A218798 Paul D. Hanna, <a href="/A218798/b218798.txt">Table of n, a(n) for n = 0..100</a>
%F A218798 a(n) = Sum_{k=1..n-1} (-1)^[(n-k-1)/2] * binomial(n,k) * k^(n-k) * a(k) for n>1 with a(0)=a(1)=1.
%e A218798 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 224*x^4/4! + 5665*x^5/5! +...
%e A218798 By definition, the coefficients a(n) satisfy:
%e A218798 1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(2*x)-sin(2*x))*x^2/2! + 15*(cos(3*x)-sin(3*x))*x^3/3! + 224*(cos(4*x)-sin(4*x))*x^4/4! + 5665*(cos(5*x)-sin(5*x))*x^5/5! +...+ a(n)*(cos(n*x)-sin(n*x))*x^n/n! +...
%o A218798 (PARI) a(n)=local(A=[1,1],N);for(i=1,n,A=concat(A,0);N=#A;A[N]=(N-1)!*(-Vec(sum(m=0,N-1,A[m+1]*x^m/m!*(cos(m*x+x*O(x^N))-sin(m*x+x*O(x^N)))))[N]));A[n+1]
%o A218798 for(n=0,25,print1(a(n),", "))
%o A218798 (PARI) a(n)=if(n<2,1,sum(k=1,n-1,(-1)^((n-k-1)\2)*a(k)*binomial(n,k)*k^(n-k)))
%o A218798 for(n=0,25,print1(a(n),", "))
%Y A218798 Cf. A219504.
%K A218798 nonn
%O A218798 0,3
%A A218798 _Paul D. Hanna_, Nov 21 2012