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Triangle read by rows: T(n,k) = ((n+k)/2)!/k! if n,k have same parity, otherwise 0.
1

%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