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

2,5

Links

Seiichi Manyama, Antidiagonals n = 2..141, flattened

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.

Example

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, ...

Prog

(PARI) T(n, k) = (-((n+1)\2)+sum(j=1, n, j*k^(n-j)))/(k+1);

Crossrefs

Columns k=0..8 give A004526, A002620, A178420, A097137, A097138, A097139, A178719, A178730, A178827.

Cf. A055129, A126885, A368343.

Sequence in context: A080242 A183927 A035317 * A103923 A349432 A186711

Adjacent sequences: A368293 A368294 A368295 * A368297 A368298 A368299

Keyword

nonn,tabl,easy

Author

Seiichi Manyama, Dec 20 2023

Status

approved