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%I #8 Feb 10 2021 01:43:18
%S 1,1,1,1,3,1,1,13,13,1,1,73,121,73,1,1,481,1081,1081,481,1,1,3601,
%T 10081,13681,10081,3601,1,1,30241,100801,171361,171361,100801,30241,1,
%U 1,282241,1088641,2217601,2782081,2217601,1088641,282241,1,1,2903041,12700801,30119041,45360001,45360001,30119041,12700801,2903041,1
%N Triangle T(n, k) = n!*binomial(n, k) - n! + 1, read by rows.
%H G. C. Greubel, <a href="/A174690/b174690.txt">Rows n = 0..100 of the triangle, flattened</a>
%F T(n, k) = n!*binomial(n, k) - n! + 1.
%F From _G. C. Greubel_, Feb 09 2021: (Start)
%F T(n, k) = A196347(n, k) - n! + 1 = (-1)^k * A021012(n, k) - n! + 1.
%F Sum_{k=0..n} T(n, k) = 2^n * n! - (n+1)! + (n+1) = A000165(n) - (n+1)! + (n+1). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 13, 13, 1;
%e 1, 73, 121, 73, 1;
%e 1, 481, 1081, 1081, 481, 1;
%e 1, 3601, 10081, 13681, 10081, 3601, 1;
%e 1, 30241, 100801, 171361, 171361, 100801, 30241, 1;
%e 1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1;
%t T[n_, k_]:= n!*Binomial[n, k] - n! + 1;
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
%o (Sage) flatten([[factorial(n)*(binomial(n,k) -1) + 1 for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 09 2021
%o (Magma) [Factorial(n)*(Binomial(n,k) -1) + 1: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 09 2021
%Y Cf. A000165, A021012, A196347.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Mar 27 2010
%E Edited by _G. C. Greubel_, Feb 09 2021
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