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).

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, 2700, 360, 0, 60, -882, 14161, 77175, 80850, 26460, 2520, 0, 0, -608, 5488, 239267, 499800, 311640, 70560, 5040, 0, -504, 6480, -57404, 735588, 3563721, 3969000, 1640520, 272160, 15120

Offset: 0

Author

Peter Luschny, Mar 14 2022

Keywords

• sign
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 0,    1;
[2] 0,    1,     2;
[3] 0,    1,     9,      6;
[4] 0,    0,    17,     36,      12;
[5] 0,   -2,    50,    325,     300,      60;
[6] 0,    0,    28,   2475,    5250,    2700,     360;
[7] 0,   60,  -882,  14161,   77175,   80850,   26460,   2520;
[8] 0,    0,  -608,   5488,  239267,  499800,  311640,  70560,  5040;
.
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).
		

Crossrefs

Cf. A027641/A027642, A048803, A007947, A264428 (Bell transform).

Programs

Mathematica

B[n_, k_] := BellY[n, k, Table[BernoulliB[j, 1], {j, 0, n}]];
P[n_] := Select[Divisors[n], PrimeQ];
T[n_, k_] := B[n, k] Product[Product[p, {p, P[j]}], {j, 1, n}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Formula

T(n, n) = Product_{k=1..n} rad(k) = Product_{k=1..n} A007947(k) = A048803(n).