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%I #10 Feb 23 2021 12:39:09
%S 1,1,1,1,-4,1,1,24,24,1,1,-138,-118,-138,1,1,1110,780,780,1110,1,1,
%T -10120,-8188,-3358,-8188,-10120,1,1,100856,101976,30240,30240,101976,
%U 100856,1,1,-1088710,-1332574,-512062,-60478,-512062,-1332574,-1088710,1
%N Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.
%H G. C. Greubel, <a href="/A169654/b169654.txt">Rows n = 1..100 of the triangle, flattened</a>
%F T(n, k) = t(n, k) + t(n, n-k+1) - t(n, 1) - t(n, n) + 1, where t(n, k) = (-1)^n*(n!/k!)*binomial(n-1, k-1).
%F T(n, k) = A008297(n,k) + A008297(n,n-k+1) - (A008297(n,1) + A008297(n,n)) + 1.
%F From _G. C. Greubel_, Feb 23 2021: (Start)
%F T(n, k) = A169653(n, k) - A169653(n, 1) + 1
%F T(n, k) = A169653(n, k) - (-1)^n * (n! + 1) + 1.
%F T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1) - (n! + 1) + (-1)^n).
%F Sum_{k=1..n} T(n, k) = (-1)^n *(2 * A000262(n) - n*(n! + 1) + (-1)^n * n). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -4, 1;
%e 1, 24, 24, 1;
%e 1, -138, -118, -138, 1;
%e 1, 1110, 780, 780, 1110, 1;
%e 1, -10120, -8188, -3358, -8188, -10120, 1;
%e 1, 100856, 101976, 30240, 30240, 101976, 100856, 1;
%e 1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710, 1;
%e 1, 12700890, 18147240, 9132480, 816480, 816480, 9132480, 18147240, 12700890, 1;
%t t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
%t T[n_, m_] = t[n, m] + t[n, n-m+1] - (-1)^n*(n! + 1) + 1;
%t Table[T[n,k], {n,12}], {k,n}]//Flatten (* modified by _G. C. Greubel_, Feb 23 2021 *)
%o (Sage)
%o def A001263(n, k): return binomial(n-1, k-1)*binomial(n,k-1)/k
%o def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
%o def A169654(n, k): return A169653(n, k) - A169653(n, 1) + 1
%o flatten([[A169654(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Feb 23 2021
%o (Magma)
%o A001263:= func< n,k | Binomial(n-1, k-1)*Binomial(n,k-1)/k >;
%o A169653:= func< n,k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
%o A169654:= func< n,k | A169653(n, k) - A169653(n, 1) + 1 >;
%o [A169654(n, k): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Feb 23 2021
%Y Cf. A000262, A001263, A008297, A105278, A169653.
%K sign,tabl,easy,less
%O 1,5
%A _Roger L. Bagula_, Apr 05 2010
%E Edited by _G. C. Greubel_, Feb 23 2021
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