%I #13 Sep 19 2018 04:20:24
%S 1,0,1,1,0,1,0,2,0,1,2,0,3,0,1,0,6,0,4,0,1,6,0,12,0,5,0,1,0,24,0,20,0,
%T 6,0,1,24,0,60,0,30,0,7,0,1,0,120,0,120,0,42,0,8,0,1,120,0,360,0,210,
%U 0,56,0,9,0,1,0,720,0,840,0,336,0,72,0,10,0,1,720,0,2520,0,1680,0,504,0,90,0,11,0,1
%N Triangle read by rows: T(n,k) = ((n+k)/2)!/k! if n,k have same parity, otherwise 0.
%H Robert Israel, <a href="/A242653/b242653.txt">Table of n, a(n) for n = 0..10010</a>
%H Alexander Kreinin, <a href="http://arxiv.org/abs/1405.5852">Combinatorial Properties of Mills' Ratio</a>, arXiv:1405.5852, 2014. See Table 4.
%e Triangle begins:
%e 1
%e 0 1
%e 1 0 1
%e 0 2 0 1
%e 2 0 3 0 1
%e 0 6 0 4 0 1
%e 6 0 12 0 5 0 1
%e 0 24 0 20 0 6 0 1
%e ...
%p N:= 1000; # to get a(0) to a(N)
%p count:= -1;
%p for n from 0 while count < N do
%p for k from 0 to n while count < N do
%p count:= count+1;
%p if type(n-k,even) then
%p A[count]:= ((n+k)/2)!/k!
%p else
%p A[count]:= 0
%p fi;
%p od
%p od:
%p seq(A[i],i=0..N); # _Robert Israel_, Jun 10 2014
%t Table[If[EvenQ[n-k], ((n+k)/2)!/k!, 0], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018 *)
%K nonn,tabl,easy
%O 0,8
%A _N. J. A. Sloane_, May 29 2014