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A172089 Triangle T(n,m) = n!/(m!!*(n-m)!!) read by rows, where (.)!! = A006882(.) are double factorials.

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