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

Offset: 0

Author

David Lovler, Jun 01 2022

Keywords

• nonn
• tabl
• easy

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.

Examples

			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, ...
  ...
		

Links

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

Crossrefs

Cf. A000290, A008794, A133728, A213037, A247375.

Cf. A266222, A266439.

Cf. A319929, A322630, A322744, A327259, A327263, A354594, A354595.

Programs

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 *)

PARI

T(n,k) = k^2 + (2*n-4)*(k\2)^2;

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).