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Triangle T(n, k) = k! - n! + (n-k)! read by rows.
3

%I #12 Feb 08 2021 05:18:40

%S 1,1,1,1,0,1,1,-3,-3,1,1,-17,-20,-17,1,1,-95,-112,-112,-95,1,1,-599,

%T -694,-708,-694,-599,1,1,-4319,-4918,-5010,-5010,-4918,-4319,1,1,

%U -35279,-39598,-40194,-40272,-40194,-39598,-35279,1,1,-322559,-357838,-362154,-362736,-362736,-362154,-357838,-322559,1

%N Triangle T(n, k) = k! - n! + (n-k)! read by rows.

%C Row sums are: 1, 2, 2, -4, -52, -412, -3292, -28492, -270412, -2810572, -31840972, ...

%H G. C. Greubel, <a href="/A155170/b155170.txt">Rows n = 0..100 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n, m) = 2*A003422(n+1) - A000142(n+1). - _R. J. Mathar_, Jun 24 2011

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 0, 1;

%e 1, -3, -3, 1;

%e 1, -17, -20, -17, 1;

%e 1, -95, -112, -112, -95, 1;

%e 1, -599, -694, -708, -694, -599, 1;

%e 1, -4319, -4918, -5010, -5010, -4918, -4319, 1;

%e 1, -35279, -39598, -40194, -40272, -40194, -39598, -35279, 1;

%e 1, -322559, -357838, -362154, -362736, -362736, -362154, -357838, -322559, 1;

%t Table[k! -n! +(n-k)!, {n,0,12}, {k, 0, n}]//Flatten

%o (Sage) f=factorial; flatten([[f(n-k) -f(n) +f(k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 07 2021

%o (Magma) F:=Factorial;; [F(n-k) -F(n) +F(k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 07 2021

%Y Cf. A000142, A003422, A176151, A176152.

%K sign,tabl,easy

%O 0,8

%A _Roger L. Bagula_, Jan 21 2009