A185958 Accumulation array of the array max{n,k}, by antidiagonals.

1, 3, 3, 6, 7, 6, 10, 13, 13, 10, 15, 21, 22, 21, 15, 21, 31, 34, 34, 31, 21, 28, 43, 49, 50, 49, 43, 28, 36, 57, 67, 70, 70, 67, 57, 36, 45, 73, 88, 94, 95, 94, 88, 73, 45, 55, 91, 112, 122, 125, 125, 122, 112, 91, 55, 66, 111, 139, 154, 160, 161, 160, 154, 139, 111, 66, 78, 133, 169, 190, 200, 203, 203, 200, 190, 169, 133, 78, 91, 157, 202, 230, 245, 251, 252, 251, 245, 230, 202, 157, 91, 105, 183, 238, 274, 295, 305, 308, 308, 305, 295, 274, 238, 183, 105

Offset: 1

Author

Clark Kimberling, Feb 07 2011

Keywords

• nonn
• tabl

Comments

A member of the accumulation chain

... < A185917 < A051125 < A185958 < ...,

where A051125, written as a rectangular array M, is given by M(n,k)=max{n,k}. See A144112 for the definition of accumulation array.

row 1: A000217

row 2: A002061

diag (1,7,...): A002412

diag (3,13,..): A016061

antidiagonal sums: A070893

Examples

			Northwest corner:
1....3....6....10....15
3....7....13...21....31
6....13...22...34....49
10...21...34...50....70
		

Crossrefs

Cf. A144112, A051125.

Programs

Maple

A := proc(n,k) option remember; ## n >= 0 and k = 0 .. n
    if k < 0 or k > n then
        0
    elif n = 0 then
        1
    else
        A(n-1,k) + A(n-1,k-1) - A(n-2,k-1) + max(n-k+1,k+1)
    end if
end proc: # Yu-Sheng Chang, Jun 05 2020

Formula

From Yu-Sheng Chang, Jun 05 2020: (Start)

O.g.f.: F(z,v) = -(v^2*z^3+v*z^3-3*v*z^2+1)/((v*z^2-v*z-z+1)^2*(v*z^2-1)*(z-1)*(v*z-1)).

T(n,k) = [v^k] 1/2*n^2*(v^(n+2)+1)/(1-v)^2+1/2*n*(3*v^(n+3)-7*v^(n+2)+7*v-3)/(-1+v)^3-1/2*v*((1-v^(1/2))^4*(-1)^n+(1+v^(1/2))^4)*v^(1/2*n)/(1-v)^4+(6*v^2+6*v^(n+2)+v^(n+4)-3*v^(n+3)-3*v+1)/(1-v)^4.

(End)