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%I #15 Jun 28 2018 02:45:14
%S 2,3,4,5,18,8,7,67,72,16,11,220,470,240,32,13,697,2625,2420,720,64,17,
%T 2100,13559,20230,10360,2016,128,19,6159,66374,152313,120400,39200,
%U 5376,256,23,17340,313136,1071168,1235346,602784,135744,13824,512,29,47581
%N 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.
%C 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).
%C Also the Bell transform of the prime numbers. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%H Alois P. Heinz, <a href="/A145520/b145520.txt">Rows n = 1..141, flattened</a>
%H Tilman Neumann, <a href="http://www.tilman-neumann.de/science.html">More terms, partitions generator and transform implementation</a>
%e Triangle begins:
%e : 2;
%e : 3, 4;
%e : 5, 18, 8;
%e : 7, 67, 72, 16;
%e : 11, 220, 470, 240, 32;
%e : 13, 697, 2625, 2420, 720, 64;
%e : 17, 2100, 13559, 20230, 10360, 2016, 128;
%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(x
%p *binomial(n-1, j-1)*ithprime(j)*b(n-j), j=1..n)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
%p seq(T(n), n=1..10); # _Alois P. Heinz_, May 27 2015
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> ithprime(n+1), 9); # _Peter Luschny_, Jan 29 2016
%t 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_ *)
%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 12;
%t B = BellMatrix[Function[n, Prime[n+1]], rows];
%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y Cf. A000040, A007446 (row sums), A145518.
%K nonn,tabl
%O 1,1
%A _Tilman Neumann_, Oct 12 2008, Oct 13 2008, Sep 02 2009
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