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A185958
Accumulation array of the array max{n,k}, by antidiagonals.
2
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
1,2
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
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)
EXAMPLE
Northwest corner:
1....3....6....10....15
3....7....13...21....31
6....13...22...34....49
10...21...34...50....70
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
CROSSREFS
Sequence in context: A021301 A016652 A008867 * A273062 A269170 A003879
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 07 2011
STATUS
approved