A178078 - OEIS

%I #9 Jun 14 2023 17:13:07

%S 1,0,1,1,4,12,42,147,527,1914,7039,26159,98110,370919,1412211,5410273,

%T 20841886,80685792,313747624,1224895416,4799435482,18867423751,

%U 74394859297,294152650731,1166021396660,4632969618849,18448290723435

%N Sequence with a (1,-1) Somos-4 Hankel transform.

%C Hankel transform is A178079.

%H G. C. Greubel, <a href="/A178078/b178078.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2306.05025">Integer sequences from elliptic curves</a>, arXiv:2306.05025 [math.NT], 2023.

%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-2*k-i)*3^(n-2*k-i)*(-2)^i*1^(k-i) ).

%t Table[Sum[(Binomial[n-k, k]/(n-2*k+1))*Sum[Binomial[k, j]*Binomial[n-k-j-1, n-2*k-j]*3^(n-2*k-j)*(-2)^j*1^(k-j), {j, 0, k}], {k, 0, Floor[n/2]}] + ((1 + (-1)^n)*(2/3)^(n/2))/2, {n, 0, 50}]  (* _G. C. Greubel_, Sep 18 2018 *)

%o (PARI) a(n) = 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)*3^(n-2*k-j)*(-2)^j));

%o for(n=0,50, print1(a(n), ", ")) \\ _G. C. Greubel_, Sep 18 2018

%K easy,nonn

%O 0,5

%A _Paul Barry_, May 19 2010