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Triangle read by rows: T(n,k) = (-1)^(n+1) * A000166(n) + (-1)^(k) * A000166(k) for n >= 2 and 1 <= k <= n-1.
1

%I #14 Aug 03 2024 19:24:02

%S -1,2,3,-9,-8,-11,44,45,42,53,-265,-264,-267,-256,-309,1854,1855,1852,

%T 1863,1810,2119,-14833,-14832,-14835,-14824,-14877,-14568,-16687,

%U 133496,133497,133494,133505,133452,133761,131642,148329,-1334961,-1334960,-1334963,-1334952,-1335005,-1334696,-1336815,-1320128,-1468457

%N Triangle read by rows: T(n,k) = (-1)^(n+1) * A000166(n) + (-1)^(k) * A000166(k) for n >= 2 and 1 <= k <= n-1.

%F Integral_{1..e} (log(x)^k - log(x)^n) dx = T(n,k)*e + A373967(n,k).

%e Triangle begins:

%e -1;

%e 2, 3;

%e -9, -8, -11;

%e 44, 45, 42, 53;

%e -265, -264, -267, -256, -309;

%e 1854, 1855, 1852, 1863, 1810, 2119;

%e ...

%t T[n_,k_]:= (-1)^(n+1)*Subfactorial[n] + (-1)^k*Subfactorial[k]; Table[T[n,k],{n,2,10},{k,n-1}]// Flatten (* _Stefano Spezia_, Jun 24 2024 *)

%Y Cf. A153805, A373967.

%Y Unsigned columns: A000166, A000240.

%Y Unsigned diagonals: A000255, A018934.

%K sign,tabl,easy

%O 2,2

%A _Mohammed Yaseen_, Jun 24 2024