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A387126 Triangle read by rows: T(n, k) = (n! / (n - k)!) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

%I A387126 #9 Aug 18 2025 18:48:51
%S A387126 1,1,1,2,4,4,6,18,36,36,12,48,144,288,288,60,300,1200,3600,7200,7200,
%T A387126 360,2160,10800,43200,129600,259200,259200,2520,17640,105840,529200,
%U A387126 2116800,6350400,12700800,12700800,5040,40320,282240,1693440,8467200,33868800,101606400,203212800,203212800
%N A387126 Triangle read by rows: T(n, k) = (n! / (n - k)!) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.
%F A387126 T(n, k) = A048803(n) * A008279(n, k).
%e A387126 Triangle begins:
%e A387126   [0]    1;
%e A387126   [1]    1,     1;
%e A387126   [2]    2,     4,      4;
%e A387126   [3]    6,    18,     36,     36;
%e A387126   [4]   12,    48,    144,    288,     288;
%e A387126   [5]   60,   300,   1200,   3600,    7200,    7200;
%e A387126   [6]  360,  2160,  10800,  43200,  129600,  259200,   259200;
%e A387126   [7] 2520, 17640, 105840, 529200, 2116800, 6350400, 12700800, 12700800;
%p A387126 A387126 := (n, k) -> mul(NumberTheory:-Radical(j), j = 1..n) * n! / (n - k)!:
%t A387126 A387126[n_, k_] := Pochhammer[n-k+1, k] Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
%t A387126 Table[A387126[n, k], {n, 0, 8}, {k, 0, n}]  // Flatten
%Y A387126 Cf. A007947 (radical), A008279, A048803 (column 0), A277174 (main diagonal).
%K A387126 nonn,tabl
%O A387126 0,4
%A A387126 _Peter Luschny_, Aug 18 2025