A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices.

1, 3, 3, 6, 18, 6, 10, 60, 60, 10, 15, 150, 300, 150, 15, 21, 315, 1050, 1050, 315, 21, 28, 588, 2940, 4900, 2940, 588, 28, 36, 1008, 7056, 17640, 17640, 7056, 1008, 36, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 55, 2475, 29700, 138600, 291060

Offset: 1

Author

Gary W. Adamson, Sep 02 2007

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle are:
1;
3, 3;
6, 18, 6;
10, 60, 60, 10;
15, 150, 300, 150, 15;
21, 315, 1050, 1050, 315, 21;
...
		

Crossrefs

Cf. A127773, A001263, A002457 (row sums), A006472. A002695, A008459.

Programs

Maple

A132818 := proc(n,k)
    (n+1-k)*binomial(n+1,k)*binomial(n,k-1)/2 ;
end proc: # R. J. Mathar, Jul 29 2015

Formula

T(n,k) = A000217(n) * A001263(n,k).

Let a(n) = A006472(n), the 'triangular' factorial numbers. Then a(n)/(a(k)*a(n-k)) produces the present triangle (with a different offset). - Peter Bala, Dec 07 2011

T(n,k) = 1/2*(n+1-k)*C(n+1,k)*C(n,k-1), for n,k >= 1. O.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(3/2) = x*y + x^2*(3*y + 3*y^2) + x^3*(6*y + 18*y^2 + 6*y^3) + .... Cf. A008459 with o.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(1/2). Sum {k = 1..n-1} T(n,k)*2^(n-k) = A002695(n). - Peter Bala, Apr 10 2012

Extensions

Corrected by R. J. Mathar, Jul 29 2015