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

1,2

Links

Table of n, a(n) for n=1..50.

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

Example

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;
...

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

Crossrefs

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

Sequence in context: A384072 A038076 A123286 * A134068 A025256 A052560

Adjacent sequences: A132815 A132816 A132817 * A132819 A132820 A132821

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Sep 02 2007

Extensions

Corrected by R. J. Mathar, Jul 29 2015

Status

approved