OFFSET
0,5
COMMENTS
The general form of this, and related triangular sequences, takes the form A(n, k, m) = (m*(n-k) + 1)*A(n-1, k-1, m) + (m*k + 1)*A(n-1, k, m) + m*f(n, k)* A(n-2, k-1, m), where f(n,k) is a polynomial in n and k.
Row sums are: 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, ... = A007590(n+1). - N. J. A. Sloane, Aug 27 2009
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..10000
FORMULA
T(n, k) = min(1 + 2*k, 1 + 2*(n - k), n).
From Yu-Sheng Chang, May 19 2020: (Start)
O.g.f.: F(z,v) = (1+v)*z/((1-v*z-1)*(1-z)*(1-v*z^2)).
T(n,k) = [v^k] (1+v)*(2*v^(n+1)+2-((sqrt(v)-1)^2 * (-1)^n + (sqrt(v)+1)^2) * v^((1/2)*n))/(2*(v-1)^2). (End)
EXAMPLE
Triangle begins as:
0;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 3, 4, 3, 1;
1, 3, 5, 5, 3, 1;
1, 3, 5, 6, 5, 3, 1;
1, 3, 5, 7, 7, 5, 3, 1;
1, 3, 5, 7, 8, 7, 5, 3, 1;
1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1;
MAPLE
T := proc(m, n) return min(1+2*m, 1+2*(n-m), n): end: seq(seq(T(m, n), m=0..n), n=0..14); # Nathaniel Johnston, Apr 29 2011
MATHEMATICA
T[n_, k_]:= Min[1+2*k, 1+2*(n-k), n]; Table[T[n, k], {n, 0, 14}, {k, 0, n}]//Flatten
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 01 2009
EXTENSIONS
Edited by N. J. A. Sloane, Aug 27 2009
More terms from and partially edited by G. C. Greubel, May 21 2020
STATUS
approved