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A242653 Triangle read by rows: T(n,k) = ((n+k)/2)!/k! if n,k have same parity, otherwise 0.

%I A242653 #16 Jun 12 2025 14:29:26
%S A242653 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 A242653 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 A242653 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 A242653 Triangle read by rows: T(n,k) = ((n+k)/2)!/k! if n,k have same parity, otherwise 0.
%H A242653 Robert Israel, <a href="/A242653/b242653.txt">Table of n, a(n) for n = 0..10010</a>
%H A242653 Alexander Kreinin, <a href="http://arxiv.org/abs/1405.5852">Combinatorial Properties of Mills' Ratio</a>, arXiv:1405.5852, 2014. See Table 4.
%e A242653 Triangle begins:
%e A242653   1
%e A242653   0  1
%e A242653   1  0  1
%e A242653   0  2  0  1
%e A242653   2  0  3  0 1
%e A242653   0  6  0  4 0 1
%e A242653   6  0 12  0 5 0 1
%e A242653   0 24  0 20 0 6 0 1
%e A242653   ...
%p A242653 N:= 1000; # to get a(0) to a(N)
%p A242653 count:= -1;
%p A242653 for n from 0 while count < N do
%p A242653   for k from 0 to n while count  < N do
%p A242653     count:= count+1;
%p A242653     if type(n-k,even) then
%p A242653        A[count]:= ((n+k)/2)!/k!
%p A242653     else
%p A242653        A[count]:= 0
%p A242653     fi;
%p A242653   od
%p A242653 od:
%p A242653 seq(A[i],i=0..N); # _Robert Israel_, Jun 10 2014
%t A242653 Table[If[EvenQ[n-k], ((n+k)/2)!/k!, 0], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018 *)
%Y A242653 Cf. A122851.
%K A242653 nonn,tabl,easy
%O A242653 0,8
%A A242653 _N. J. A. Sloane_, May 29 2014