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%I #33 Aug 12 2022 09:17:47
%S 3,-6,10,-12,3,34,-114,204,-114,-636,2676,-5528,3939,17778,-83994,
%T 186972,-150438,-609524,3091020,-7204008,6237902,23649204,-125807412,
%U 302476536,-275144388,-996903096,5489607272,-13498689840,12721569699,44596212754,-252074322858
%N Expansion of ((1 + 4*x + 8*x^2)^(3/2) - (1 + 6*x + 18*x^2 + 20*x^3)) / (2*x^4) in powers of x.
%H Ewan Delanoy, <a href="https://math.stackexchange.com/q/2728009/">Divisibility property for sequence a(n+2) = -2(n-1)(n+3)a(n) - (2n+3)a(n+1)</a>, Mathematics Stack Exchange question 2728009, Apr 08 2018.
%F 0 = (8*n + 8)*a(n) + (4*n + 14)*a(n+1) + (n + 6)*a(n+2) for all n in Z if a(-1)=10, a(-2)=9, a(-3)=3, a(-4)=1/2, and also
%F 0 = a(n)*(+64*a(n+1) +112*a(n+2) +48*a(n+3)) +a(n+1)*(-48*a(n+1) -16*a(n+2) +14*a(n+3)) +a(n+2)*(-6*a(n+2) +a(n+3)) for all n in Z.
%F D-finite with recurrence (n+4)*a(n) +2*(2*n+3)*a(n-1) +8*(n-1)*a(n-2)=0. - _R. J. Mathar_, Sep 24 2021
%e G.f. = 3 - 6*x + 10*x^2 - 12*x^3 + 3*x^4 + 34*x^5 - 114*x^6 + 204*x^7 + ...
%t a[ n_] := If[ n < 1, 3 Boole[n==0], Sum[ (-1)^k Binomial[k, 2 k - n - 4] (2 k - 5)! / (2^(k - 3) k! (k - 3)!), {k, 3, n + 4}] 24 2^n];
%o (PARI) {a(n) = if( n<1, 3*(n==0), sum(k=3, n+4, (-1)^k * binomial(k, 2*k-n-4) * (2*k-5)! / (2^(k-3) * k! * (k-3)!)) * 24 * 2^n)};
%K sign
%O 0,1
%A _Michael Somos_, Oct 26 2018
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