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%I #16 Sep 08 2022 08:45:50
%S 1,1,1,1,2,1,2,3,3,2,3,8,6,8,3,8,15,20,20,15,8,15,48,45,80,45,48,15,
%T 48,105,168,210,210,168,105,48,105,384,420,896,630,896,420,384,105,
%U 384,945,1728,2520,3024,3024,2520,1728,945,384,945,3840,4725,11520,9450,16128,9450,11520,4725,3840,945
%N Triangle T(n,m) = n!/(m!!*(n-m)!!) read by rows, where (.)!! = A006882(.) are double factorials.
%C Row sums are {1, 2, 4, 10, 28, 86, 296, 1062, 4240, 17202, 77088, ...}.
%H G. C. Greubel, <a href="/A172089/b172089.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,m) = A000142(n)/(A006882(m)*A006882(n-m)).
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 2, 3, 3, 2;
%e 3, 8, 6, 8, 3;
%e 8, 15, 20, 20, 15, 8;
%e 15, 48, 45, 80, 45, 48, 15;
%e 48, 105, 168, 210, 210, 168, 105, 48;
%e 105, 384, 420, 896, 630, 896, 420, 384, 105;
%e 384, 945, 1728, 2520, 3024, 3024, 2520, 1728, 945, 384;
%e 945, 3840, 4725, 11520, 9450, 16128, 9450, 11520, 4725, 3840, 945;
%p A172089 := proc(n,m)
%p factorial(n)/doublefactorial(m)/doublefactorial(n-m) ;
%p end proc:
%p seq(seq(A172089(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Oct 11 2011
%t binomialn[n_, k_] = n!/(Factorial2[n-k]*Factorial2[k]); Table[binomialn[n, k], {n,0,10}, {k,0,n}]//Flatten
%o (PARI)
%o f2(n) = prod(i=0, (n-1)\2, n - 2*i );
%o T(n,k) = n!/(f2(k)*f2(n-k));
%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 05 2019
%o (Magma)
%o F2:=func< n | &*[n..2 by -2] >;
%o [Factorial(n)/(F2(k)*F2(n-k)): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 05 2019
%o (Sage)
%o def T(n, k): return factorial(n)/((k).multifactorial(2)*(n-k).multifactorial(2))
%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 05 2019
%Y Cf. A000142, A006882.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Jan 25 2010
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