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A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows.

%I A169653 #10 Feb 23 2021 02:32:15
%S A169653 -2,3,3,-7,-12,-7,25,48,48,25,-121,-260,-240,-260,-121,721,1830,1500,
%T A169653 1500,1830,721,-5041,-15162,-13230,-8400,-13230,-15162,-5041,40321,
%U A169653 141176,142296,70560,70560,142296,141176,40321,-362881,-1451592,-1695456,-874944,-423360,-874944,-1695456,-1451592,-362881
%N A169653 Triangle T(n,k) = A008297(n,k) + A008297(n,n-k+1), read by rows.
%H A169653 G. C. Greubel, <a href="/A169653/b169653.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A169653 T(n, k) = t(n, k) + t(n, n-k+1), where t(n, k) = (-1)^n*(n!/k!)*binomial(n-1, k-1).
%F A169653 T(n, k) = A008297(n,k) + A008297(n,n-k+1).
%F A169653 From _G. C. Greubel_, Feb 23 2021: (Start)
%F A169653 T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1)).
%F A169653 T(n, k) = (-1)^n * ( k! + (n-k+1)! ) * A001263(n, k).
%F A169653 Sum_{k=1..n} T(n, k) = 2 * (-1)^n * A000262(n). (End)
%e A169653 Triangle begins as:
%e A169653      -2;
%e A169653       3,      3;
%e A169653      -7,    -12,     -7;
%e A169653      25,     48,     48,    25;
%e A169653    -121,   -260,   -240,  -260,   -121;
%e A169653     721,   1830,   1500,  1500,   1830,    721;
%e A169653   -5041, -15162, -13230, -8400, -13230, -15162,  -5041;
%e A169653   40321, 141176, 142296, 70560,  70560, 142296, 141176, 40321;
%t A169653 t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
%t A169653 T[n_, m_] = t[n, m] + t[n, n-m+1];
%t A169653 Table[T[n,k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Feb 23 2021 *)
%o A169653 (Sage)
%o A169653 def A001263(n, k): return binomial(n-1, k-1)*binomial(n,k-1)/k
%o A169653 def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
%o A169653 flatten([[A169653(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Feb 23 2021
%o A169653 (Magma)
%o A169653 A001263:= func< n,k | Binomial(n-1, k-1)*Binomial(n,k-1)/k >;
%o A169653 A169653:= func< n,k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
%o A169653 [A169653(n, k): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Feb 23 2021
%Y A169653 Cf. A000262, A001263, A008297, A105278, A169154.
%K A169653 sign,tabl
%O A169653 1,1
%A A169653 _Roger L. Bagula_, Apr 05 2010
%E A169653 Edited by _G. C. Greubel_, Feb 23 2021