OFFSET
1,1
COMMENTS
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.
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.
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).
2nd column (4, 6, 19, 29, 68, 94, 177, ...) is A024697.
Row sums give A145519.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Tilman Neumann, More terms, partition generator and transform implementation.
EXAMPLE
Triangle starts:
2;
3, 4;
5, 6, 8;
7, 19, 12, 16;
11, 29, 38, 24, 32;
13, 68, 85, 76, 48, 64;
...
MAPLE
g:= proc(n, i) option remember; `if`(n=0 or i=1, (2*x)^n,
expand(add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2)):
seq(T(n), n=1..12); # Alois P. Heinz, May 25 2015
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tilman Neumann, Oct 12 2008
EXTENSIONS
Reference to more terms etc. changed to make it version independent by Tilman Neumann, Sep 02 2009
STATUS
approved