A145520 Triangle read by rows: T2[n,k] = Sum_{partitions of n with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} c(n; m_1, m_2, ..., m_n) * x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i = i-th prime.

2, 3, 4, 5, 18, 8, 7, 67, 72, 16, 11, 220, 470, 240, 32, 13, 697, 2625, 2420, 720, 64, 17, 2100, 13559, 20230, 10360, 2016, 128, 19, 6159, 66374, 152313, 120400, 39200, 5376, 256, 23, 17340, 313136, 1071168, 1235346, 602784, 135744, 13824, 512, 29, 47581

Offset: 1

Author

Tilman Neumann, Oct 12 2008, Oct 13 2008, Sep 02 2009

Keywords

• nonn
• tabl

Comments

Here c(n; m_1, m_2, ..., m_n) = n! / (m_1!*1!^m_1 * m_2!*2!^m_2 * ... * m_n!*n!^m_n) is the number of ways to realize the partition p(n, k; m_1, m_2, m_3, ..., m_n).

Also the Bell transform of the prime numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Examples

			Triangle begins:
:  2;
:  3,    4;
:  5,   18,     8;
:  7,   67,    72,    16;
: 11,  220,   470,   240,    32;
: 13,  697,  2625,  2420,   720,   64;
: 17, 2100, 13559, 20230, 10360, 2016, 128;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Tilman Neumann, More terms, partitions generator and transform implementation

Crossrefs

Cf. A000040, A007446 (row sums), A145518.

Programs

Maple

b:= proc(n) option remember; expand(`if`(n=0, 1, add(x
      *binomial(n-1, j-1)*ithprime(j)*b(n-j), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 27 2015
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> ithprime(n+1), 9); # Peter Luschny, Jan 29 2016

Mathematica

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[x*Binomial[n - 1, j - 1]*Prime[j]* b[n - j], {j, 1, n}]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Prime[n+1]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)