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

1, 3, 3, 6, 10, 6, 10, 21, 21, 10, 15, 36, 46, 36, 15, 21, 55, 81, 81, 55, 21, 28, 78, 126, 146, 126, 78, 28, 36, 105, 181, 231, 231, 181, 105, 36, 45, 136, 246, 336, 371, 336, 246, 136, 45, 55, 171, 321, 461, 546, 546, 461, 321, 171, 55, 66, 210, 406, 606, 756, 812, 756

Offset: 1

Author

Clark Kimberling, Feb 07 2011

Keywords

• nonn
• tabl

Comments

A member of the accumulation chain

... < A003982 < A003783 < A115262 < A185957 <...,

where A003783(n,k)=min{n,k}. See A144112 for the definition of accumulation array.

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.

Examples

			Northwest corner:
1....3....6....10...15
3....10...21...36...55
6....21...46...81...126
10...36...81...146..231
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A144112, A003783, A115262.

Programs

Mathematica

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k (k + 1)/2, {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A000292 *)
Table[m[1, j], {j, 1, 12}] (* A000217 *)
Table[m[2, j], {j, 1, 12}] (* A014105 *)
Table[m[j, j], {j, 1, 12}] (* A024166 *)
Table[m[j, j + 1], {j, 1, 12}] (* A112851 *)
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}] (* A001769 *)