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A352372 Triangle read by rows. Let R(n, k) = Y(n, k, B) where Y are the partial Bell polynomials and B is the list [Bernoulli(j, 1), j = 0..n]. T(n, k) are R(n, k) normalized by the lcm of the denominators of the terms in row n (A048803).

%I A352372 #8 Mar 14 2022 17:36:15
%S A352372 1,0,1,0,1,2,0,1,9,6,0,0,17,36,12,0,-2,50,325,300,60,0,0,28,2475,5250,
%T A352372 2700,360,0,60,-882,14161,77175,80850,26460,2520,0,0,-608,5488,239267,
%U A352372 499800,311640,70560,5040,0,-504,6480,-57404,735588,3563721,3969000,1640520,272160,15120
%N A352372 Triangle read by rows. Let R(n, k) = Y(n, k, B) where Y are the partial Bell polynomials and B is the list [Bernoulli(j, 1), j = 0..n]. T(n, k) are R(n, k) normalized by the lcm of the denominators of the terms in row n (A048803).
%F A352372 T(n, n) = Product_{k=1..n} rad(k) = Product_{k=1..n} A007947(k) = A048803(n).
%e A352372 Triangle starts:
%e A352372 [0] 1;
%e A352372 [1] 0,    1;
%e A352372 [2] 0,    1,     2;
%e A352372 [3] 0,    1,     9,      6;
%e A352372 [4] 0,    0,    17,     36,      12;
%e A352372 [5] 0,   -2,    50,    325,     300,      60;
%e A352372 [6] 0,    0,    28,   2475,    5250,    2700,     360;
%e A352372 [7] 0,   60,  -882,  14161,   77175,   80850,   26460,   2520;
%e A352372 [8] 0,    0,  -608,   5488,  239267,  499800,  311640,  70560,  5040;
%e A352372 .
%e A352372 For example row 7 is 2520*[R(7, k), k = 0..7] = 2520*[0, 1/42, -7/20, 2023/360, 245/8, 385/12, 21/2, 1] since lcm(1, 42, 20, 360, 8, 12, 2, 1) = A048803(7) = 2520. Conversely, since R(n, n) = 1 and T(n, n) = Product_{k=1..n} rad(k), the R(n, k) can be obtained by dividing the terms of row n by T(n, n).
%t A352372 B[n_, k_] := BellY[n, k, Table[BernoulliB[j, 1], {j, 0, n}]];
%t A352372 P[n_] := Select[Divisors[n], PrimeQ];
%t A352372 T[n_, k_] := B[n, k] Product[Product[p, {p, P[j]}], {j, 1, n}];
%t A352372 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%Y A352372 Cf. A027641/A027642, A048803, A007947, A264428 (Bell transform).
%K A352372 sign,tabl
%O A352372 0,6
%A A352372 _Peter Luschny_, Mar 14 2022