A140356 Triangle T(n,m) read by rows: m! if m <= floor(n/2), and (n-m)! otherwise.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6, 24, 6, 2, 1, 1, 1, 1, 2, 6, 24, 24, 6, 2, 1, 1, 1, 1, 2, 6, 24, 120, 24, 6, 2, 1, 1, 1, 1, 2, 6, 24, 120, 120, 24, 6, 2, 1, 1, 1, 1, 2, 6, 24, 120, 720, 120, 24, 6, 2, 1, 1

Offset

0,13

Comments

Row sums: 1, 2, 3, 4, 6, 8, 14, 20, 44, 68, 188,... which is

2*A003422((n+1)/2) if n is odd, and A003422(n/2)+A003422(1+n/2) if n is even.

Links

Table of n, a(n) for n=0..90.

Formula

T(n,m) = A000142(m) if m<=[n/2], = A000142(n-m) if m>[n/2], 0<=m<=n.

Conjecture: limit_{n->infinity} sum_{m=0..n} ( 1/binomial(n,m) - T(n,m) ) = 0.

T(n,m) = T(n,n-m).

T(2n,n) = n! = A000142(n).

Example

  1,
  1, 1,
  1, 1, 1,
  1, 1, 1, 1,
  1, 1, 2, 1,  1,
  1, 1, 2, 2,  1,   1,
  1, 1, 2, 6,  2,   1,  1,
  1, 1, 2, 6,  6,   2,  1, 1,
  1, 1, 2, 6, 24,   6,  2, 1, 1,
  1, 1, 2, 6, 24,  24,  6, 2, 1, 1,
  1, 1, 2, 6, 24, 120, 24, 6, 2, 1, 1,
  ...

Maple

T:= (n, k) -> min(k, n-k)!:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Mar 21 2026

Mathematica

g[n_, m_] := If[m <= Floor[n/2], m!, (n - m)! ]; w = Table[Table[g[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[w] Table[Apply[Plus, Table[g[n, m], {m, 0, n}]], {n, 0, 10}]

Crossrefs

Cf. A000142, A107044.

Sequence in context: A139147 A055801 A155050 * A119963 A057790 A350889

Adjacent sequences: A140353 A140354 A140355 * A140357 A140358 A140359

Keyword

nonn,easy,tabl

Author

Roger L. Bagula and Gary W. Adamson, May 30 2008

Extensions

Non-Ascii characters corrected, offset set to 0, reported Mma experiments removed - The Assoc. Editors of the OEIS, Oct 31 2009

Status

approved