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 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
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 LINKS Alois P. Heinz, Rows n = 1..141, flattened Tilman Neumann, More terms, partitions generator and transform implementation EXAMPLE 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; 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 *) CROSSREFS Cf. A000040, A007446 (row sums), A145518. Sequence in context: A162657 A305794 A317944 * A061412 A295317 A165646 Adjacent sequences:  A145517 A145518 A145519 * A145521 A145522 A145523 KEYWORD nonn,tabl AUTHOR Tilman Neumann, Oct 12 2008, Oct 13 2008, Sep 02 2009 STATUS approved

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Last modified December 15 16:15 EST 2018. Contains 318150 sequences. (Running on oeis4.)