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A387138 Triangle read by rows: T(n, k) = binomial(n, k) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.

%I A387138 #9 Aug 18 2025 18:49:00
%S A387138 1,1,1,2,4,2,6,18,18,6,12,48,72,48,12,60,300,600,600,300,60,360,2160,
%T A387138 5400,7200,5400,2160,360,2520,17640,52920,88200,88200,52920,17640,
%U A387138 2520,5040,40320,141120,282240,352800,282240,141120,40320,5040
%N A387138 Triangle read by rows: T(n, k) = binomial(n, k) * Product_{k=1..n} radical(k), where radical(n) is the product of distinct prime factors of n, cf. A007947.
%F A387138 T(n, k) = A048803(n) * A007318(n, k).
%e A387138 Triangle begins:
%e A387138   [0]    1;
%e A387138   [1]    1,     1;
%e A387138   [2]    2,     4,      2;
%e A387138   [3]    6,    18,     18,      6;
%e A387138   [4]   12,    48,     72,     48,     12;
%e A387138   [5]   60,   300,    600,    600,    300,     60;
%e A387138   [6]  360,  2160,   5400,   7200,   5400,   2160,    360;
%e A387138   [7] 2520, 17640,  52920,  88200,  88200,  52920,  17640,  2520;
%e A387138   [8] 5040, 40320, 141120, 282240, 352800, 282240, 141120, 40320, 5040;
%p A387138 A387138 := (n, k) -> binomial(n, k) * mul(NumberTheory:-Radical(j), j = 1..n):
%t A387138 A387138[n_, k_] := Binomial[n, k] Times @@ ResourceFunction["IntegerRadical"][Range[1, n]];
%t A387138 Table[A387138[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A387138 Cf. A007318 (binomial), A007947 (radical), A048803 (column 0 and main diagonal), A387139 (row sums), A387126.
%K A387138 nonn,tabl
%O A387138 0,4
%A A387138 _Peter Luschny_, Aug 18 2025