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A172088 Triangle: T(n,m) = n!! - m!! - (n-m)!! read by rows 0 <= m <= n, where ()!! are the double factorials.

%I A172088 #21 Jun 10 2025 23:34:16
%S A172088 -1,-1,-1,-1,0,-1,-1,0,0,-1,-1,4,4,4,-1,-1,6,10,10,6,-1,-1,32,38,42,
%T A172088 38,32,-1,-1,56,88,94,94,88,56,-1,-1,278,334,366,368,366,334,278,-1,
%U A172088 -1,560,838,894,922,922,894,838,560,-1,-1,2894,3454,3732,3784,3810,3784,3732,3454,2894,-1
%N A172088 Triangle: T(n,m) = n!! - m!! - (n-m)!! read by rows 0 <= m <= n, where ()!! are the double factorials.
%C A172088 Row sums are {-1, -2, -2, -2, 10, 30, 180, 474, 2322, 6426, 31536, ...}; n-th row sum is (n+1)*n!! - 2*A129981(n).
%H A172088 G. C. Greubel, <a href="/A172088/b172088.txt">Rows n = 0..100 of triangle, flattened</a>
%F A172088 T(n,m) = A006882(n) - A006882(m) - A006882(n-m).
%e A172088 Triangle begins
%e A172088   -1;
%e A172088   -1,   -1;
%e A172088   -1,    0,   -1;
%e A172088   -1,    0,    0,   -1;
%e A172088   -1,    4,    4,    4,   -1;
%e A172088   -1,    6,   10,   10,    6,   -1;
%e A172088   -1,   32,   38,   42,   38,   32,   -1;
%e A172088   -1,   56,   88,   94,   94,   88,   56,   -1;
%e A172088   -1,  278,  334,  366,  368,  366,  334,  278,   -1;
%e A172088   -1,  560,  838,  894,  922,  922,  894,  838,  560,   -1;
%e A172088   -1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1;
%p A172088 A172088 := proc(n,m)
%p A172088         doublefactorial(n)-doublefactorial(m)-doublefactorial(n-m) ;
%p A172088 end proc:
%p A172088 seq(seq(A172088(n,m),m=0..n),n=0..10) ; # _R. J. Mathar_, Oct 11 2011
%t A172088 T[n_, k_] = n!! -k!! -(n-k)!!; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A172088 (PARI) f2(n) = prod(j=0, (n-1)\2, n-2*j);
%o A172088 T(n,k) = f2(n) - f2(k) - f2(n-k); \\ _G. C. Greubel_, Dec 05 2019
%o A172088 (Magma) F2:=func< n | &*[n..2 by -2] >;
%o A172088 [F2(n) - F2(k) - F2(n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 05 2019
%o A172088 (Sage)
%o A172088 def T(n, k): return (n).multifactorial(2) - (k).multifactorial(2) - (n-k).multifactorial(2)
%o A172088 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 05 2019
%Y A172088 Cf. A006882, A129981.
%K A172088 sign,tabl,easy
%O A172088 0,12
%A A172088 _Roger L. Bagula_, Jan 25 2010