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

1,2

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A144112, A003783, A115262.

Sequence in context: A274498 A167786 A167787 * A124326 A202970 A205004

Adjacent sequences: A185954 A185955 A185956 * A185958 A185959 A185960

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 07 2011

Status

approved