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 A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices. 3

%I

%S 1,3,3,6,18,6,10,60,60,10,15,150,300,150,15,21,315,1050,1050,315,21,

%T 28,588,2940,4900,2940,588,28,36,1008,7056,17640,17640,7056,1008,36,

%U 45,1620,15120,52920,79380,52920,15120,1620,45,55,2475,29700,138600,291060

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

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

%F 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

%F 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

%e First few rows of the triangle are:

%e 1;

%e 3, 3;

%e 6, 18, 6;

%e 10, 60, 60, 10;

%e 15, 150, 300, 150, 15;

%e 21, 315, 1050, 1050, 315, 21;

%e ...

%p A132818 := proc(n,k)

%p (n+1-k)*binomial(n+1,k)*binomial(n,k-1)/2 ;

%p end proc: # _R. J. Mathar_, Jul 29 2015

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

%K nonn,tabl,easy

%O 1,2

%A _Gary W. Adamson_, Sep 02 2007

%E Corrected by _R. J. Mathar_, Jul 29 2015

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Last modified November 27 15:50 EST 2020. Contains 338683 sequences. (Running on oeis4.)