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).

2, 3, 4, 5, 6, 8, 7, 19, 12, 16, 11, 29, 38, 24, 32, 13, 68, 85, 76, 48, 64, 17, 94, 181, 170, 152, 96, 128, 19, 177, 326, 443, 340, 304, 192, 256, 23, 231, 683, 787, 886, 680, 608, 384, 512, 29, 400, 1066, 1780, 1817, 1772, 1360, 1216, 768, 1024, 31, 484, 1899, 3119

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

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

Sequence in context: A299442 A299440 A330401 * A256221 A359099 A210253

Adjacent sequences: A145515 A145516 A145517 * A145519 A145520 A145521

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