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A185784 Accumulation array of A107985, by antidiagonals.

%I A185784 #10 Jul 13 2017 03:07:53
%S A185784 1,4,4,10,15,10,20,36,36,20,35,70,84,70,35,56,120,160,160,120,56,84,
%T A185784 189,270,300,270,189,84,120,280,420,500,500,420,280,120,165,396,616,
%U A185784 770,825,770,616,396,165,220,540,864,1120,1260,1260,1120,864,540,220,286,715,1170,1560,1820,1911,1820,1560,1170,715,286,364,924,1540,2100,2520,2744,2744,2520,2100,1540,924,364,455,1170,1980,2750,3375,3780,3920,3780,3375,2750,1980,1170,455,560
%N A185784 Accumulation array of A107985, by antidiagonals.
%C A185784 Let W be the array given by w(1,1)=1, w(2,2)=-1, and w(n,k)=0 for all other (n,k).
%C A185784 Write "A < B" to indicate that an array B is the accumulation array of A (defined at A144112). Then  W < A103451 < A002024 < A107985 < A185784 < A185785 < A185786.
%H A185784 G. C. Greubel, <a href="/A185784/b185784.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185784 T(n,k) = (n+k+1)*C(n+1,2)*C(k+1,2)/3, k>=0, n>=0.
%e A185784 Northwest corner:
%e A185784 1....4....10....20....35
%e A185784 4....15...36....70....120
%e A185784 10...36...84....160...270
%e A185784 20...70...160...300...500
%t A185784 (* The code generates arrays A107985, A185784, A002024. *)
%t A185784 f[n_,0]:=0;f[0,k_]:=0; (* used to form A002024 *)
%t A185784 f[n_,k_]:=k*n(k+n)/2;
%t A185784 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A107985 *)
%t A185784 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185784 FullSimplify[s[n,k]]
%t A185784 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A185784 *)
%t A185784 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185784 w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
%t A185784 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A002024 *)
%Y A185784 Cf. A144112, A103451, A002024, A107985, A185784, A185785, A185786.
%K A185784 nonn,tabl
%O A185784 1,2
%A A185784 _Clark Kimberling_, Feb 03 2011