cp's OEIS Frontend

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

%I A185958 #16 Sep 04 2020 15:40:10
%S A185958 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,
%T A185958 49,43,28,36,57,67,70,70,67,57,36,45,73,88,94,95,94,88,73,45,55,91,
%U A185958 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
%N A185958 Accumulation array of the array max{n,k}, by antidiagonals.
%C A185958 A member of the accumulation chain
%C A185958 ... < A185917 < A051125 < A185958 < ...,
%C 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.
%C A185958 row 1: A000217
%C A185958 row 2: A002061
%C A185958 diag (1,7,...): A002412
%C A185958 diag (3,13,..): A016061
%C A185958 antidiagonal sums: A070893
%F A185958 From _Yu-Sheng Chang_, Jun 05 2020: (Start)
%F A185958 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)).
%F A185958 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.
%F A185958 (End)
%e A185958 Northwest corner:
%e A185958 1....3....6....10....15
%e A185958 3....7....13...21....31
%e A185958 6....13...22...34....49
%e A185958 10...21...34...50....70
%p A185958 A := proc(n,k) option remember; ## n >= 0 and k = 0 .. n
%p A185958     if k < 0 or k > n then
%p A185958         0
%p A185958     elif n = 0 then
%p A185958         1
%p A185958     else
%p A185958         A(n-1,k) + A(n-1,k-1) - A(n-2,k-1) + max(n-k+1,k+1)
%p A185958     end if
%p A185958 end proc: # _Yu-Sheng Chang_, Jun 05 2020
%Y A185958 Cf. A144112, A051125.
%K A185958 nonn,tabl
%O A185958 1,2
%A A185958 _Clark Kimberling_, Feb 07 2011