A003982 -id:A003982 - 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 *)

A286100 Square array A(n,k): If n = k, then A(n,k) = n, otherwise 0, read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, May 03 2017

			The top left 9 X 9 corner of the array:
  1, 0, 0, 0, 0, 0, 0, 0, 0
  0, 2, 0, 0, 0, 0, 0, 0, 0
  0, 0, 3, 0, 0, 0, 0, 0, 0
  0, 0, 0, 4, 0, 0, 0, 0, 0
  0, 0, 0, 0, 5, 0, 0, 0, 0
  0, 0, 0, 0, 0, 6, 0, 0, 0
  0, 0, 0, 0, 0, 0, 7, 0, 0
  0, 0, 0, 0, 0, 0, 0, 8, 0
  0, 0, 0, 0, 0, 0, 0, 0, 9

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array

Cf. A000027 (the main diagonal).

Cf. also arrays A003982, A285732.

Table[Function[s, If[OddQ@ Length@ s, ReplacePart[s, {# -> #}] &[Ceiling[n/2]], s]]@ ConstantArray[0, n], {n, 15}] // Flatten (* Michael De Vlieger, May 04 2017 *)

def A(n, k): return n if n==k else 0
for n in range(1, 21): print( [A(k, n - k + 1) for k in range(1, n + 1)] ) # Indranil Ghosh, May 03 2017

(define (A286100 n) (A286100bi (A002260 n) (A004736 n)))
(define (A286100bi row col) (if (= row col) row 0))

If n = k, then A(n,k) = n, otherwise 0.

cp's OEIS Frontend

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

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