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%I #12 Apr 22 2024 12:37:41
%S 1,2,0,4,-8,24,-72,224,-720,2368,-7936,27008,-93088,324288,-1140032,
%T 4039296,-14409728,51713792,-186577152,676334592,-2462090752,
%U 8997154816,-32992079872,121362092032,-447721572864,1656081763328,-6140640246784,22820403312640
%N G.f. A(x) satisfies A(x) = ( 1 + 4*x*(1 + x*A(x)) )^(1/2).
%F G.f.: A(x) = (1+4*x)/(-2*x^2 + sqrt(1+4*x+4*x^4)).
%F a(n) = Sum_{k=0..n} 4^k * binomial(n/2-k/2+1/2,k) * binomial(k,n-k)/(n-k+1).
%F D-finite with recurrence n*a(n) +2*(2*n-3)*a(n-1) +4*(n-6)*a(n-4)=0. - _R. J. Mathar_, Apr 22 2024
%p A372002 := proc(n)
%p add(4^k*binomial((n-k+1)/2,k)*binomial(k,n-k)/(n-k+1),k=0..n) ;
%p end proc:
%p seq(A372002(n),n=0..60) ; # _R. J. Mathar_, Apr 22 2024
%o (PARI) my(N=30, x='x+O('x^N)); Vec((1+4*x)/(-2*x^2+sqrt(1+4*x+4*x^4)))
%o (PARI) a(n) = sum(k=0, n, 4^k*binomial(n/2-k/2+1/2, k)*binomial(k, n-k)/(n-k+1));
%Y Cf. A372003.
%K sign
%O 0,2
%A _Seiichi Manyama_, Apr 15 2024
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