Voozh

A354596

Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

0, 1, 0, 0, 1, 0, 5, 2, 1, 0, 0, 7, 4, 1, 0, 9, 8, 9, 6, 1, 0, 0, 17, 16, 11, 8, 1, 0, 13, 18, 25, 24, 13, 10, 1, 0, 0, 31, 36, 33, 32, 15, 12, 1, 0, 17, 32, 49, 54, 41, 40, 17, 14, 1, 0, 0, 49, 64, 67, 72, 49, 48, 19, 16, 1, 0, 21, 50, 81, 96, 85, 90, 57, 56, 21, 18, 1, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

Column k is an arithmetic progression with difference 2*A008794(k).

Odd rows of A133728 triangle are contained in row 0.

For i = 0 through 4, row i is 0 and the diagonal of A319929, A322630 = A213037, A003991, A322744, and A327259, respectively. In general, row i is 0 and the diagonal of array U(i;n,k) described in A327263.

LINKS

David Lovler, Table of n, a(n) for n = 0..5150

FORMULA

T(n,k) = U(n;k,k) (see A327263).

For each row, T(n,k) = T(n,k-1) + 2*T(n,k-2) - 2*T(n,k-3) - T(n,k-4) + T(n,k-5), k >= 5.

G.f. for row n: x*(1 + (2*n-1)*x + 3*x^2 + (2*n-3)*x^3)/((1 - x)^3*(1 + x)^2). When n = 2, this reduces to x*(1 + x)/(1 - x)^3.

E.g.f. for row n: (((4-n)*x + n*x^2)*cosh(x) + (n-2 + n*x + n*x^2)*sinh(x))/2. When n = 2, this reduces to (x + x^2)*cosh(x) + (x + x^2)*sinh(x) = (x + x^2)*exp(x).

EXAMPLE

T(n,k) begins:

0, 1, 0, 5, 0, 9, 0, 13, ...

0, 1, 2, 7, 8, 17, 18, 31, ...

0, 1, 4, 9, 16, 25, 36, 49, ...

0, 1, 6, 11, 24, 33, 54, 67, ...

0, 1, 8, 13, 32, 41, 72, 85, ...

0, 1, 10, 15, 40, 49, 90, 103, ...

0, 1, 12, 17, 48, 57, 108, 121, ...

...

MATHEMATICA

T[n_, k_] := k^2 + (2*n - 4)*Floor[k/2]^2; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jun 20 2022 *)