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A185915 Accumulation array of A185914, by antidiagonals.

%I A185915 #12 Jul 23 2017 03:54:09
%S A185915 1,3,1,6,4,1,10,9,4,1,15,16,10,4,1,21,25,19,10,4,1,28,36,31,20,10,4,1,
%T A185915 36,49,46,34,20,10,4,1,45,64,64,52,35,20,10,4,1,55,81,85,74,55,35,20,
%U A185915 10,4,1,66,100,109,100,80,56,35,20,10,4,1,78,121,136,130,110,83,56,35,20,10,4,1,91,144,166,164,145,116,84,56,35,20,10,4,1,105,169,199,202,185,155,119,84,56,35,20,10,4,1
%N A185915 Accumulation array of A185914, by antidiagonals.
%C A185915 A member of the accumulation chain ... < A185916 < A185914 < A185915 < ...
%C A185915 (See A144112 for definitions of weight array and accumulation array.)
%H A185915 G. C. Greubel, <a href="/A185915/b185915.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185915 T(n,k) = C(k+2,3) if k<=n; T(n,k) = k*(k+2-n)/2 if k>n; k>=1, n>=1.
%e A185915 Northwest corner:
%e A185915 1....3....6....10....15....21....28
%e A185915 1....4....9....16....25....36....49
%e A185915 1....4....10...19....31....46....64
%e A185915 1....4....10...20....34....52....74
%e A185915 1....4....10...20....35....55....80
%e A185915 1....4....10...20....35....56....83
%e A185915 row 1: A000217 (triangular numbers)
%e A185915 row 2: A000290 (squares)
%e A185915 row 3: A005448 (centered triangular numbers)
%e A185915 row 4: A005893
%e A185915 row 5: A062985
%e A185915 Limit of rows: A000292 (tetrahedral numbers)
%t A185915 f[n_, 0] := 0; f[0, k_] := 0; f[n_, k_] := k - n + 1; f[n_, k_] := 0 /; k < n; s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}]; Table[s[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten
%Y A185915 Cf. A144112, A185914, A185916.
%K A185915 nonn,tabl
%O A185915 1,2
%A A185915 _Clark Kimberling_, Feb 06 2011