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A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.

%I A185506 #24 Jul 11 2020 13:38:46
%S A185506 1,3,4,7,11,10,14,23,26,20,25,42,51,50,35,41,70,88,94,85,56,63,109,
%T A185506 140,156,155,133,84,92,161,210,240,250,237,196,120,129,228,301,350,
%U A185506 375,374,343,276,165,175,312,416,490,535,550,532,476,375,220,231,415,558,664,735,771,770,728,639,495,286
%N A185506 Accumulation array, T, of the natural number array A000027, by antidiagonals.
%C A185506 Suppose that R={R(n,k) : n>=1, k>=1} is a rectangular array.  The accumulation array of R is given by T(n,k) = Sum_{R(i,j): 1<=i<=n, 1<=j<=k}. (See A144112.)
%C A185506 The formula for the integer T(n,k) has denominator 12.  The 2nd, 3rd, and 4th accumulation arrays of A000027 have formulas in which the denominators are 144, 2880, and 86400, respectively; see A185507, A185508, and A185509.
%H A185506 G. C. Greubel, <a href="/A185506/b185506.txt">Table of n, a(n) for the first 50 antidiagonals, flattened</a>
%F A185506 T(n,k) = k*n*(2*n^2 + 3*(k+1)*n + 2*k^2 - 3*k + 5)/12.
%e A185506 The natural number array A000027 starts with
%e A185506   1, 2,  4,  7, ...
%e A185506   3, 5,  8, 12, ...
%e A185506   6, 9, 13, 18, ...
%e A185506   ...
%e A185506 T(n,k) is the sum of numbers in the rectangle with corners at (1,1) and (n,k) of A000027, so that a corner of T is as follows:
%e A185506    1,  3,   7,  14,  25,  41
%e A185506    4, 11,  23,  42,  70, 109
%e A185506   10, 26,  51,  88, 140, 210
%e A185506   20, 50,  94, 156, 240, 350
%e A185506   35, 85, 155, 250, 375, 535
%t A185506 f[n_,k_]:=k*n*(2n^2+3(k+1)*n+2k^2-3k+5)/12;
%t A185506 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
%t A185506 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185506 Cf. A000027, A185507, A185508, A185509.
%Y A185506 Cf. A004006 (row 1), A000292 (col 1), A051925 (col 2), A185505 (1st diagonal).
%K A185506 nonn,tabl
%O A185506 1,2
%A A185506 _Clark Kimberling_, Jan 29 2011