%I #9 Nov 02 2025 22:45:24

%S 1,0,2,-2,8,-20,64,-200,656,-2184,7408,-25456,88480,-310464,1098304,

%T -3912896,14026752,-50557184,183110400,-666079872,2432399104,

%U -8914099840,32772922112,-120844493056,446793362944,-1656004554752

%N Sequence with a (4,-8) Somos-4 Hankel transform.

%C Hankel transform is A178077.

%H G. C. Greubel, <a href="/A178076/b178076.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/2)} ( (C(n-k,k)/(n-2k+1))*Sum_{i=0..k} C(k,i)*C(n-k-i-1,n-2k-i)*(-2)^(n-2k-i)*2^i*2^(k-i) ).

%t Table[If[n == 0, 1, Sum[(Binomial[n-k,k]/(n-2*k+1))* Sum[Binomial[k, j]*Binomial[n-k-j-1, n-2*k-j]*(-2)^(n-2*k-j)*2^k, {j, 0, k}], {k, 0, Floor[n/2]}] + (1 + (-1)^n)/2], {n, 0, 100}] (* _G. C. Greubel_, Sep 18 2018 *)

%o (PARI) a(n) = if(n==0, 1, sum(k=0, floor(n/2), sum(j=0, k, (binomial(n-k, k)/(n-2*k+1)) *binomial(k, j)*binomial(n-k-j-1, n-2*k-j)*(-2)^(n-2*k-j)*2^k))+(1+(-1)^n)/2);

%o for(n=0,30, print1(a(n), ", ")) \\ _G. C. Greubel_, Sep 18 2018; corrected by Georg Fischer_, Nov 02 2025

%K easy,sign

%O 0,3

%A _Paul Barry_, May 19 2010