A104713 - OEIS

%I #14 Aug 08 2015 21:32:05

%S 1,4,1,10,5,1,20,15,6,1,35,35,21,7,1,56,70,56,28,8,1,84,126,126,84,36,

%T 9,1,120,210,252,210,120,45,10,1,165,330,462,462,330,165,55,11,1,220,

%U 495,792,924,792,495,220,66,12,1,286,715,1287,1716

%N Triangle T(n,k) = binomial(n,k), read by rows, 3 <= k <=n .

%F T(n,k) = A007318(n,k) for n>=3, 3<=k<=n.

%F From _Peter Bala_, Jul 16 2013: (Start)

%F The following remarks assume an offset of 0.

%F Riordan array (1/(1 - x)^4, x/(1 - x)).

%F O.g.f.: 1/(1 - t)^3*1/(1 - (1 + x)*t) = 1 + (4 + x)*t + (10 + 5*x + x^2)*t^2 + ....

%F E.g.f.: (1/x*d/dt)^3 (exp(t)*(exp(x*t) - 1 - x*t - (x*t)^2/2!)) = 1 + (4 + x)*t + (10 + 5*x + x^2)*t^2/2! + ....

%F The infinitesimal generator for this triangle has the sequence [4,5,6,...] on the main subdiagonal and 0's elsewhere. (End)

%e First few rows of the triangle are:

%e 1;

%e 4, 1;

%e 10, 5, 1;

%e 20, 15, 6, 1;

%e 35, 35, 21, 7, 1;

%e 56, 70, 56, 28, 8, 1;

%e ...

%p A104713 := proc(n,k)

%p         binomial(n,k) ;

%p end proc;

%p seq(seq( A104713(n,k),k=3..n),n=3..16) ; # _R. J. Mathar_, Oct 29 2011

%Y Cf. A007318, A104712, A002662 (row sums).

%K nonn,tabl,easy

%O 3,2

%A _Gary W. Adamson_, Mar 19 2005