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A185734 Third accumulation array of the polygonal number array (A086270), by antidiagonals.

%I A185734 #12 Jul 12 2017 03:18:22
%S A185734 1,6,4,21,25,10,56,90,65,20,126,245,240,135,35,252,560,665,510,245,56,
%T A185734 462,1134,1540,1435,945,406,84,792,2100,3150,3360,2695,1596,630,120,
%U A185734 1287,3630,5880,6930,6370,4606,2520,930,165,2002,5940,10230
%N A185734 Third accumulation array of the polygonal number array (A086270), by antidiagonals.
%C A185734 The chain of accumulation arrays is A144257->A086270->A185732->A185733->A184734.
%H A185734 G. C. Greubel, <a href="/A185734/b185734.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185734 T(n,k) = C(k+3,4)*C(n+2,3)*(k*n-n+3*k+17)/20, k>=1, n>=1.
%F A185734 T(n,k) = Sum_{j=1..n} Sum_{l=1..k} A185733(j,l), by definition.
%e A185734 Northwest corner:
%e A185734 1....6......21.....56....126
%e A185734 4....25.....90....245....560
%e A185734 10...65....240....665...1540
%e A185734 20...135...510...1435...3360
%t A185734 f[n_,k_]:= k*(1+k)*(2+k)*n*(1+n)*(10+2*k-n+k*n)/144;
%t A185734 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* array A185733 *)
%t A185734 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A185733 *)
%t A185734 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185734 FullSimplify[s[n,k]]  (* the formula for A185734 *)
%t A185734 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* array A185734 *)
%t A185734 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A185734 *)
%Y A185734 Cf. A144112, A144257, A086270, A185732, A185733.
%K A185734 nonn,tabl,easy
%O A185734 1,2
%A A185734 _Clark Kimberling_, Feb 02 2011