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%I #12 Jul 18 2023 02:35:58
%S 1,1,2,27,1160,113385,21060816,6623049027,3256046222848,
%T 2359986757857297,2411094849547390720,3350982557290104443883,
%U 6155509617679334624756736,14592373629282306879174535161,43755759571493116198207431532544,163135210694347619479784565520981395
%N E.g.f.: Sum_{n>=0} a(n) * (cos(n^2*x) - sin(n^2*x)) * x^n/n! = 1 + x.
%H Seiichi Manyama, <a href="/A221535/b221535.txt">Table of n, a(n) for n = 0..179</a>
%F a(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 By definition, the coefficients a(n) satisfy:
%e 1+x = 1 + 1*(cos(x)-sin(x))*x + 2*(cos(4*x)-sin(4*x))*x^2/2! + 27*(cos(9*x)-sin(9*x))*x^3/3! + 1160*(cos(16*x)-sin(16*x))*x^4/4! + 113385*(cos(25*x)-sin(25*x))*x^5/5! +...+ a(n)*(cos(n^2*x)-sin(n^2*x))*x^n/n! +...
%o (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^2*x+x*O(x^N))-sin(m^2*x+x*O(x^N)))))[N])); A[n+1]
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) a(n)=if(n<2, 1, sum(k=1, n-1, (-1)^((n-k-1)\2)*a(k)*binomial(n, k)*k^(2*n-2*k)))
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A218798, A195736, A209316, A221534.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 19 2013