A178078 Sequence with a (1,-1) Somos-4 Hankel transform.

1, 0, 1, 1, 4, 12, 42, 147, 527, 1914, 7039, 26159, 98110, 370919, 1412211, 5410273, 20841886, 80685792, 313747624, 1224895416, 4799435482, 18867423751, 74394859297, 294152650731, 1166021396660, 4632969618849, 18448290723435

Offset: 0

Author

Paul Barry, May 19 2010

Keywords

• easy
• nonn

Comments

Hankel transform is A178079.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.

Programs

Mathematica

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 *)

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));
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, Sep 18 2018

Formula

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) ).