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Odd-index Fibonacci row-sum array: T(n,k)=U(2n+1,n+1+k), 0<=k<=n, n >= 0, U the array in A054125.
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%I #21 May 30 2022 23:36:21

%S 2,3,2,6,5,2,12,13,7,2,24,30,24,9,2,48,65,68,39,11,2,96,136,171,134,

%T 58,13,2,192,279,398,394,236,81,15,2,384,566,880,1040,802,382,108,17,

%U 2,768,1141,1880,2542,2396,1479,580,139,19,2,1536

%N Odd-index Fibonacci row-sum array: T(n,k)=U(2n+1,n+1+k), 0<=k<=n, n >= 0, U the array in A054125.

%F From _Jianing Song_, May 30 2022: (Start)

%F T(n,k) = 2 if k = n, otherwise A052509(2n,n+1+k) + A052509(2n,n-k) = 2^(n-1-k) + Sum_{m=0..n-k} binomial(n+k,m) = 2^(n-1-k) + 2^(n+k) - Sum_{m=0..2*k-1} binomial(n+k,m).

%F T(n,k) = [x^n*y^(n-k)] (1-x*y) * ((1+y-x*y^2)/((1-x*y^2)*((1-x*y)^2-x)) + (1+y-x*y)/((1-x)*((1-x*y)^2-x*y^2))). (End)

%e Rows:

%e 2;

%e 3, 2;

%e 6, 5, 2;

%e 12, 13, 7, 2;

%e ...

%o (PARI) T(n,k) = if(k==n, 2, 2^(n-1-k) + sum(m=0, n-k, binomial(n+k, m))) \\ _Jianing Song_, May 30 2022

%K nonn,tabl,eigen

%O 0,1

%A _Clark Kimberling_