OFFSET
0,7
COMMENTS
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 *)
PROG
(PARI) T(n, k) = k^2 + (2*n-4)*(k\2)^2;
CROSSREFS
KEYWORD
AUTHOR
David Lovler, Jun 01 2022
STATUS
approved