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A209316 E.g.f.: Sum_{n>=0} a(n) * (cos(n^2*x) - sin(n^2*x)) * x^n/n! = 1/(1-x).

%I A209316 #14 Jan 24 2016 16:06:43
%S A209316 1,1,4,57,2456,240205,44616096,14030856525,6897867308800,
%T A209316 4999592004999705,5107861266649227520,7098997630368216900833,
%U A209316 13040338287878632604362752,30913685990004537377333201253,92695803952674372198927320920064,345599063527286969179932122231749365
%N A209316 E.g.f.: Sum_{n>=0} a(n) * (cos(n^2*x) - sin(n^2*x)) * x^n/n!  =  1/(1-x).
%F A209316 a(n) = n! + Sum_{k=1..n-1} (-1)^[(n-k-1)/2] * binomial(n,k) * k^(2*n-2*k) * a(k) for n>1 with a(0)=a(1)=1.
%e A209316 By definition, the coefficients a(n) satisfy:
%e A209316 1/(1-x) = 1 + 1*(cos(x)-sin(x))*x + 4*(cos(4*x)-sin(4*x))*x^2/2! + 57*(cos(9*x)-sin(9*x))*x^3/3! + 2456*(cos(16*x)-sin(16*x))*x^4/4! + 240205*(cos(25*x)-sin(25*x))*x^5/5! +...+ a(n)*(cos(n^2*x)-sin(n^2*x))*x^n/n! +...
%o A209316 (PARI) a(n)=local(A=[1, 1], N); for(i=1, n, A=concat(A, 0); N=#A; A[N]=(N-1)!*(1-Vec(sum(m=0, N-1, A[m+1]*x^m/m!*(cos(m^2*x+x*O(x^N))-sin(m^2*x+x*O(x^N)))))[N])); A[n+1]
%o A209316 for(n=0, 25, print1(a(n), ", "))
%o A209316 (PARI) a(n)=if(n==0 || n==1, 1, n!+sum(k=1, n-1, (-1)^((n-k-1)\2)*a(k)*binomial(n, k)*k^(2*n-2*k)))
%o A209316 for(n=0, 25, print1(a(n), ", "))
%Y A209316 Cf. A219504, A221535, A220282, A209317.
%K A209316 nonn
%O A209316 0,3
%A A209316 _Paul D. Hanna_, Jan 19 2013