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A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.

%I A169654 #14 Jun 10 2025 23:34:01
%S A169654 1,1,1,1,-4,1,1,24,24,1,1,-138,-118,-138,1,1,1110,780,780,1110,1,1,
%T A169654 -10120,-8188,-3358,-8188,-10120,1,1,100856,101976,30240,30240,101976,
%U A169654 100856,1,1,-1088710,-1332574,-512062,-60478,-512062,-1332574,-1088710,1
%N A169654 Triangle T(n, k) = A169643(n, k) - A169653(n, 1) + 1, read by rows.
%H A169654 G. C. Greubel, <a href="/A169654/b169654.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A169654 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 A169654 T(n, k) = A008297(n,k) + A008297(n,n-k+1) - (A008297(n,1) + A008297(n,n)) + 1.
%F A169654 From _G. C. Greubel_, Feb 23 2021: (Start)
%F A169654 T(n, k) = A169653(n, k) - A169653(n, 1) + 1
%F A169654 T(n, k) = A169653(n, k) - (-1)^n * (n! + 1) + 1.
%F A169654 T(n, k) = (-1)^n * (A105278(n, k) + A105278(n, n-k+1) - (n! + 1) + (-1)^n).
%F A169654 Sum_{k=1..n} T(n, k) = (-1)^n *(2 * A000262(n) - n*(n! + 1) + (-1)^n * n). (End)
%e A169654 Triangle begins as:
%e A169654   1;
%e A169654   1,        1;
%e A169654   1,       -4,        1;
%e A169654   1,       24,       24,       1;
%e A169654   1,     -138,     -118,    -138,      1;
%e A169654   1,     1110,      780,     780,   1110,       1;
%e A169654   1,   -10120,    -8188,   -3358,  -8188,  -10120,        1;
%e A169654   1,   100856,   101976,   30240,  30240,  101976,   100856,        1;
%e A169654   1, -1088710, -1332574, -512062, -60478, -512062, -1332574, -1088710,        1;
%e A169654   1, 12700890, 18147240, 9132480, 816480,  816480,  9132480, 18147240, 12700890, 1;
%t A169654 t[n_, m_] = (-1)^n*(n!/m!)*Binomial[n-1, m-1];
%t A169654 T[n_, m_] = t[n, m] + t[n, n-m+1] - (-1)^n*(n! + 1) + 1;
%t A169654 Table[T[n,k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Feb 23 2021 *)
%o A169654 (Sage)
%o A169654 def A001263(n, k): return binomial(n-1, k-1)*binomial(n,k-1)/k
%o A169654 def A169653(n, k): return (-1)^n*A001263(n, k)*(factorial(k) + factorial(n-k+1))
%o A169654 def A169654(n, k): return A169653(n, k) - A169653(n, 1) + 1
%o A169654 flatten([[A169654(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Feb 23 2021
%o A169654 (Magma)
%o A169654 A001263:= func< n,k | Binomial(n-1, k-1)*Binomial(n,k-1)/k >;
%o A169654 A169653:= func< n,k | (-1)^n*A001263(n, k)*(Factorial(k) + Factorial(n-k+1)) >;
%o A169654 A169654:= func< n,k | A169653(n, k) - A169653(n, 1) + 1 >;
%o A169654 [A169654(n, k): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Feb 23 2021
%Y A169654 Cf. A000262, A001263, A008297, A105278, A169653.
%K A169654 sign,tabl,easy,less
%O A169654 1,5
%A A169654 _Roger L. Bagula_, Apr 05 2010
%E A169654 Edited by _G. C. Greubel_, Feb 23 2021