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%I #23 Jan 03 2022 21:12:45
%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;
%e ...
%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-François 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 Reference to more terms etc. changed to make it version independent by _Tilman Neumann_, Sep 02 2009
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