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A145518 Triangle read by rows: T1[n,k;x] := sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i). 5

%I

%S 2,3,4,5,6,8,7,19,12,16,11,29,38,24,32,13,68,85,76,48,64,17,94,181,

%T 170,152,96,128,19,177,326,443,340,304,192,256,23,231,683,787,886,680,

%U 608,384,512,29,400,1066,1780,1817,1772,1360,1216,768,1024,31,484,1899,3119

%N Triangle read by rows: T1[n,k;x] := sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).

%C Let p(n; m_1, m_2, m_3, ..., m_n) denote a partition of integer n in exponential representation, i.e. the m_i are the counts of parts i and satisfy 1*m_1 + 2*m_2 + 3*m_3 + ... + n*m_n = n.

%C Let p(n, k; m_1, m_2, m_3, ..., m_n) be the partitions of n into exactly k parts; these are further constrained by m_1 + m_2 + m_3 + ... + m_n = k.

%C Then the triangle is given by T1[n,k;x] := sum_{all p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i is the i-th prime number (A000040).

%C 2nd column (4,6,19,29,68,94,177, ...) is A024697.

%C Row sums give A145519.

%H Alois P. Heinz, <a href="/A145518/b145518.txt">Rows n = 1..141, flattened</a>

%H Tilman Neumann, <a href="http://www.tilman-neumann.de/science.html">More terms, partition generator and transform implementation</a>.

%e Triangle starts:

%e 2;

%e 3, 4;

%e 5, 6, 8;

%e 7, 19, 12, 16;

%e 11, 29, 38, 24, 32;

%e 13, 68, 85, 76, 48, 64;

%p g:= proc(n, i) option remember; `if`(n=0 or i=1, (2*x)^n,

%p expand(add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..n/i)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2)):

%p seq(T(n), n=1..12); # _Alois P. Heinz_, May 25 2015

%t g[n_, i_] := g[n, i] = If[n==0 || i==1, (2 x)^n, Expand[Sum[g[n-i*j, i-1]*(Prime[i]*x)^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n]]; Table[T[n], {n, 1, 12}] // Flatten (* _Jean-Fran├žois Alcover_, Jul 15 2015, after _Alois P. Heinz_ *)

%Y Cf. A000040, A024697, A145519, A145520, A258323.

%K nonn,tabl

%O 1,1

%A _Tilman Neumann_, Oct 12 2008

%E Changed reference to more terms etc. to make it version independent _Tilman Neumann_, Sep 02 2009

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Last modified March 27 12:35 EDT 2017. Contains 284176 sequences.