A054126 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.
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, 58, 13, 2, 192, 279, 398, 394, 236, 81, 15, 2, 384, 566, 880, 1040, 802, 382, 108, 17, 2, 768, 1141, 1880, 2542, 2396, 1479, 580, 139, 19, 2, 1536
Offset: 0
Examples
Rows: 2; 3, 2; 6, 5, 2; 12, 13, 7, 2; ...
Programs
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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
Formula
From Jianing Song, May 30 2022: (Start)
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).
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)