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A185731 Array by antidiagonals: T(n,k)=F*(k*n-n+3*k+13), where F = k*(k+1)*(k+2)*n*(n+1)*(n+2)/576.

%I A185731 #8 Jul 12 2017 03:17:54
%S A185731 1,5,4,15,21,10,35,65,55,20,70,155,175,115,35,126,315,425,375,210,56,
%T A185731 210,574,875,925,700,350,84,330,966,1610,1925,1750,1190,546,120,495,
%U A185731 1530,2730,3570,3675,3010,1890,810,165,715,2310,4350,6090,6860,6370,4830,2850,1155,220,1001,3355,6600
%N A185731 Array by antidiagonals: T(n,k)=F*(k*n-n+3*k+13), where F = k*(k+1)*(k+2)*n*(n+1)*(n+2)/576.
%C A185731 This is the accumulation array of A185730.  (See A144112 for the definition of accumulation array.)
%H A185731 G. C. Greubel, <a href="/A185731/b185731.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185731 T(n,k) = F*(k*n-n+3*k+13), where F = k*(k+1)*(k+2)*n*(n+1)*(n+2)/576.
%e A185731 Northwest corner:
%e A185731 1.....5....15....35....70
%e A185731 4.....21...65....155...315
%e A185731 10....55...175...425...875
%e A185731 20....115..375...925...1925
%t A185731 f[n_,k_]:=k(1+k)n(1+n)(7+2k-n+k*n)/36;
%t A185731 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* A185730 *)
%t A185731 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185731 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
%t A185731 Factor[s[n,k]]  (* formula for A185731 *)
%t A185731 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* array A185731 *)
%t A185731 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185731 *)
%Y A185731 Cf. A144112, A185730.
%K A185731 nonn,tabl
%O A185731 1,2
%A A185731 _Clark Kimberling_, Feb 01 2011