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A185957 Second accumulation array of the array min{n,k}, by antidiagonals.

%I A185957 #13 Jul 23 2017 22:03:49
%S A185957 1,3,3,6,10,6,10,21,21,10,15,36,46,36,15,21,55,81,81,55,21,28,78,126,
%T A185957 146,126,78,28,36,105,181,231,231,181,105,36,45,136,246,336,371,336,
%U A185957 246,136,45,55,171,321,461,546,546,461,321,171,55,66,210,406,606,756,812,756
%N A185957 Second accumulation array of the array min{n,k}, by antidiagonals.
%C A185957 A member of the accumulation chain
%C A185957 ... < A003982 < A003783 < A115262 < A185957 <...,
%C A185957 where A003783(n,k)=min{n,k}.  See A144112 for the definition of accumulation array.
%C A185957 A185957 also gives the symmetric matrix based on the triangular numbers s=(1,3,6,10,15,....; viz, let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A185957 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202678 for characteristic polynomials of principal submatrices of M.
%H A185957 G. C. Greubel, <a href="/A185957/b185957.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%e A185957 Northwest corner:
%e A185957 1....3....6....10...15
%e A185957 3....10...21...36...55
%e A185957 6....21...46...81...126
%e A185957 10...36...81...146..231
%t A185957 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k (k + 1)/2, {k, 1, 15}]];
%t A185957 L = Transpose[U]; M = L.U; TableForm[M]
%t A185957 m[i_, j_] := M[[i]][[j]];
%t A185957 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A185957 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A185957 Table[f[n], {n, 1, 12}]
%t A185957 Table[Sqrt[f[n]], {n, 1, 12}] (* A000292 *)
%t A185957 Table[m[1, j], {j, 1, 12}] (* A000217 *)
%t A185957 Table[m[2, j], {j, 1, 12}] (* A014105 *)
%t A185957 Table[m[j, j], {j, 1, 12}] (* A024166 *)
%t A185957 Table[m[j, j + 1], {j, 1, 12}] (* A112851 *)
%t A185957 Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}] (* A001769 *)
%Y A185957 Cf. A144112, A003783, A115262.
%K A185957 nonn,tabl
%O A185957 1,2
%A A185957 _Clark Kimberling_, Feb 07 2011