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%I #12 Feb 08 2021 05:17:41
%S 1,1,1,1,-10,1,1,72,72,1,1,-528,-678,-528,1,1,4770,6780,6780,4770,1,1,
%T -48025,-87568,-68458,-87568,-48025,1,1,524384,1287776,947520,947520,
%U 1287776,524384,1,1,-6169282,-19982590,-18010942,-10305790,-18010942,-19982590,-6169282,1
%N Triangle T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1) read by rows.
%C Row sums are: 1, 2, -8, 146, -1732, 23102, -339642, 5519362, -98631416, 1926628022, ...
%H G. C. Greubel, <a href="/A176021/b176021.txt">Rows n = 1..100 of the triangle, flattened</a>
%F T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1).
%F T(n, k) = 1 - (-1)^n*(n! + 1) + binomial(n+1, k)*( A008297(n, k) + A008297(n, n-k+1) )/(n+1). - _G. C. Greubel_, Feb 08 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -10, 1;
%e 1, 72, 72, 1;
%e 1, -528, -678, -528, 1;
%e 1, 4770, 6780, 6780, 4770, 1;
%e 1, -48025, -87568, -68458, -87568, -48025, 1;
%e 1, 524384, 1287776, 947520, 947520, 1287776, 524384, 1;
%e 1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1;
%t A176013[n_, k_]:= (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
%t T[n_, m_]:= 1 - (-1)^n*(n! + 1) + A176013[n, k] + A176013[n, n-k+1];
%t Table[T[n, k], {n, 12}, {k, n}]//Flatten
%o (Sage)
%o def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
%o flatten([[1 - (-1)^n*(factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 08 2021
%o (Magma)
%o A176013:= func< n,k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
%o [1 - (-1)^n*(Factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 08 2021
%Y Cf. A008297, A176013, A176022.
%K sign,tabl,easy,less
%O 1,5
%A _Roger L. Bagula_, Apr 06 2010
%E Edited by _G. C. Greubel_, Feb 08 2021
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