A112851 -id:A112851 - cp's OEIS frontend

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

Clark Kimberling, Feb 07 2011

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.

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

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

Cf. A144112, A003783, A115262.

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 *)

A112852 Table with row lengths 1 1 2 3 5 9 17 33 65 ... which counts the objects described in A047970 and A112508.

Original entry on oeis.org

1, 2, 4, 1, 7, 6, 1, 11, 20, 8, 3, 1, 16, 50, 33, 21, 10, 3, 6, 4, 1, 22, 105, 98, 81, 49, 21, 48, 36, 12, 3, 6, 12, 10, 4, 10, 5, 1, 29, 196, 238, 231, 168, 81, 210, 168, 68

Offset: 0

Alford Arnold, Sep 24 2005

The row sums are 1 2 5 14 43 144 523 2048 8597 ... A047970. The columns are essentially A000124, A002415, A051836, A112851, A051947, ...

			The table begins
1
2
4 1
7 6 1
11 20 8 3 1
16 50 33 21 10 3 6 4 1
22 105 98 81 49 21

Cf. A000124, A002415, A051836, A112851, A051947, A047970, A112508.

cp's OEIS Frontend

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

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