login
Number triangle T(n,k)=binomial(2(n+k),4k).
0

%I #5 Mar 30 2019 21:32:44

%S 1,1,1,1,15,1,1,70,45,1,1,210,495,91,1,1,495,3003,1820,153,1,1,1001,

%T 12870,18564,4845,231,1,1,1820,43758,125970,74613,10626,325,1,1,3060,

%U 125970,646646,735471,230230,20475,435,1,1,4845,319770,2704156,5311735

%N Number triangle T(n,k)=binomial(2(n+k),4k).

%C Related to matchings of the complete graph K_2n: T(n,k)=A100861(2(n+k),2k)/f(2k), where f(n)=(2n-1)!! Column k gives number of standard tableaux of shape (2n+1,1^(4k)).

%F Column k has g.f. x^k*sum{j=0..2k+1, binomial(4k+1, 2j)x^j}/(1-x)^(4k+1)

%e Rows begin

%e 1;

%e 1,1;

%e 1,15,1;

%e 1,70,45,1;

%e 1,210,495,91,1;

%t Table[Binomial[2(n+k),4k],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Mar 30 2019 *)

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Aug 17 2005