A368296 Square array T(n,k), n >= 2, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/2).
1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 8, 6, 3, 1, 5, 14, 18, 9, 3, 1, 6, 22, 44, 39, 12, 4, 1, 7, 32, 90, 135, 81, 16, 4, 1, 8, 44, 162, 363, 408, 166, 20, 5, 1, 9, 58, 266, 813, 1455, 1228, 336, 25, 5, 1, 10, 74, 408, 1599, 4068, 5824, 3688, 677, 30, 6
Offset: 2
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, 7, ... 2, 4, 8, 14, 22, 32, 44, ... 2, 6, 18, 44, 90, 162, 266, ... 3, 9, 39, 135, 363, 813, 1599, ... 3, 12, 81, 408, 1455, 4068, 9597, ... 4, 16, 166, 1228, 5824, 20344, 57586, ...
Links
- Seiichi Manyama, Antidiagonals n = 2..141, flattened
Crossrefs
Programs
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PARI
T(n, k) = (-((n+1)\2)+sum(j=1, n, j*k^(n-j)))/(k+1);
Formula
T(n,k) = T(n-2,k) + Sum_{j=0..n-2} k^j.
T(n,k) = 1/(k+1) * (-floor((n+1)/2) + Sum_{j=1..n} j*k^(n-j)).
T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k+1)) = Sum_{j=0..n} floor(k^j/(k^2-1)) for k > 1.
T(n,k) = (k+1)*T(n-1,k) - (k-1)*T(n-2,k) - (k+1)*T(n-3,k) + k*T(n-4,k).
G.f. of column k: x^2/((1-x) * (1-k*x) * (1-x^2)).
T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^2-1)) - floor((n+1)/2)) for k > 1.