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A373967 Triangle read by rows: T(n,k) = (-1)^n * n! + (-1)^(k+1) * k! for n >= 2 and 1 <= k <= n-1.

%I A373967 #17 Aug 03 2024 19:24:17
%S A373967 3,-5,-8,25,22,30,-119,-122,-114,-144,721,718,726,696,840,-5039,-5042,
%T A373967 -5034,-5064,-4920,-5760,40321,40318,40326,40296,40440,39600,45360,
%U A373967 -362879,-362882,-362874,-362904,-362760,-363600,-357840,-403200,3628801,3628798,3628806,3628776,3628920,3628080,3633840,3588480,3991680
%N A373967 Triangle read by rows: T(n,k) = (-1)^n * n! + (-1)^(k+1) * k! for n >= 2 and 1 <= k <= n-1.
%F A373967 Integral_{1..e} (log(x)^k - log(x)^n) dx = A373966(n,k)*e + T(n,k).
%e A373967 Triangle begins:
%e A373967       3;
%e A373967      -5,    -8;
%e A373967      25,    22,    30;
%e A373967    -119,  -122,  -114,  -144;
%e A373967     721,   718,   726,   696,   840;
%e A373967   -5039, -5042, -5034, -5064, -4920, -5760;
%e A373967   ...
%t A373967 T[n_,k_]:= (-1)^n*n! + (-1)^(k+1)*k!; Table[T[n,k],{n,2,10},{k,n-1}]// Flatten (* _Stefano Spezia_, Jun 24 2024 *)
%Y A373967 Cf. A000142, A153805, A373966.
%Y A373967 Unsigned diagonals: A001048, A213167.
%K A373967 sign,tabl,easy
%O A373967 2,1
%A A373967 _Mohammed Yaseen_, Jun 24 2024