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%I #50 Feb 19 2022 07:37:02
%S 1,1,0,2,2,1,6,12,12,8,24,72,108,108,81,120,480,960,1280,1280,1024,
%T 720,3600,9000,15000,18750,18750,15625,5040,30240,90720,181440,272160,
%U 326592,326592,279936,40320,282240,987840,2304960,4033680,5647152,6588344,6588344,5764801
%N Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.
%C Rows n >= 2 are coefficients in a double summation power series for the integral of x^(1/x), and the integral of its inverse function y^(y^(y^(y^(...)))). See A350358.
%F T(n, k) = binomial(n, k)*A350269(n, k). - _Peter Luschny_, Dec 25 2021
%F T(n+1, k) = A061711(n) * (n+1) / A350149(n, k). - _Robert B Fowler_, Jan 11 2022
%e Triangle T(n,k) begins:
%e -----------------------------------------------------------------
%e n\k 0 1 2 3 4 5 6 7
%e -----------------------------------------------------------------
%e 0 | 1,
%e 1 | 1, 0,
%e 2 | 2, 2, 1,
%e 3 | 6, 12, 12, 8,
%e 4 | 24, 72, 108, 108, 81,
%e 5 | 120, 480, 960, 1280, 1280, 1024,
%e 6 | 720, 3600, 9000, 15000, 18750, 18750, 15625,
%e 7 | 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936.
%e ...
%p T := (n, k) -> (n!/k!)*(n - 1)^k:
%p seq(seq(T(n, k), k = 0..n), n = 0..8); # _Peter Luschny_, Dec 24 2021
%t T[1, 0] := 1; T[n_, k_] := n!*(n - 1)^k/k!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Dec 24 2021 *)
%Y Cf. A000142 (first column), A062119 (second column), A065440 (main diagonal), A055897 (subdiagonal), A217701 (row sums).
%Y Cf. A350269, A350358.
%Y Cf. A061711, A350149.
%K easy,nonn,tabl
%O 0,4
%A _Robert B Fowler_, Dec 23 2021