A144480 T(n,k) = binomial(n, k)*min(k + 1, n - k + 1), triangle read by rows (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 80, 45, 12, 1, 1, 14, 63, 140, 140, 63, 14, 1, 1, 16, 84, 224, 350, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1

Offset: 0

Author

Roger L. Bagula, Oct 11 2008

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,    1;
  1, 10,  30,  30,   10,    1;
  1, 12,  45,  80,   45,   12,    1;
  1, 14,  63, 140,  140,   63,   14,   1;
  1, 16,  84, 224,  350,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...
		

Crossrefs

Row sums are in A245560.

Cf. A003983, A007318, A168643.

Programs

Mathematica

Table[Table[Binomial[n, m]*If[m <= Floor[n/2], 1 + m, 1 + n - m], {m, 0, n}], {n, 0, 10}] // Flatten

Maxima

create_list(binomial(n, k)*min(k + 1, n - k + 1), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Dec 10 2018 */

Formula

If k <= floor(n/2), then T(n,k) = binomial(n, k)*(k + 1), otherwise T(n,k) = binomial(n, k)*(n - k - 1).

T(n,k) = A007318(n,k)*A003983(k+1,n-k+1), i.e., term-by term product of Pascal's triangle A007318 and A003983 as a triangle.

Extensions

Entry revised by N. J. A. Sloane, Aug 07 2014

Edited by Franck Maminirina Ramaharo, Dec 10 2018