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A185880 Second accumulation array of A185877, by antidiagonals.

%I A185880 #18 Feb 13 2018 02:49:21
%S A185880 1,5,3,16,17,6,40,56,38,10,85,140,128,70,15,161,295,320,240,115,21,
%T A185880 280,553,670,600,400,175,28,456,952,1246,1250,1000,616,252,36,705,
%U A185880 1536,2128,2310,2075,1540,896,348,45,1045,2355,3408,3920,3815,3185,2240,1248,465,55,1496,3465,5190,6240,6440,5831,4620,3120,1680,605,66,2080,4928,7590,9450,10200,9800,8428,6420,4200,2200
%N A185880 Second accumulation array of A185877, by antidiagonals.
%C A185880 A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... See A144112 for the definition of accumulation array.
%H A185880 G. C. Greubel, <a href="/A185880/b185880.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185880 T(n,k) = C(k,2)*C(n,2)*(3*k^2+4*k*n-k-4*n+16)/18, k>=1, n>=1.
%e A185880 Northwest corner:
%e A185880    1,    5,   16,   40,   85
%e A185880    3,   17,   56,  140,  295
%e A185880    6,   38,  128,  320,  670
%e A185880   10,   70,  240,  600, 1250
%t A185880 (* This program generates A185878 first and then generates A185880 as the accumulation array of A185878. *)
%t A185880 f[n_,k_]:=(k*n/6)(7-3k+2k^2-3n+3kn);
%t A185880 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185878 *)
%t A185880 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185880 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
%t A185880 FullSimplify[s[n,k]]
%t A185880 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185880 *)
%t A185880 f[n_, k_] := (1/72)*k*(1 + k)*n*(1 + n)*(16 - k + 3 *k^2 + 4 *(-1 + k) *n); Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* _G. C. Greubel_, Jul 21 2017 *)
%Y A185880 Cf. A144112, A185877, A185878.
%Y A185880 Antidiagonal sums: A037235.
%Y A185880 diag (1,5,...): A056108 (4th spoke on hexagonal wheel);
%Y A185880 diag (3,11,...): A056106 (2nd spoke on hexagonal wheel);
%Y A185880 diag (7,19,...): A003215 (hex numbers);
%Y A185880 diag (13,29,...): A144391.
%K A185880 nonn,tabl
%O A185880 1,2
%A A185880 _Clark Kimberling_, Feb 05 2011