A321752 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in p(u), where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

1, 1, -2, 1, 0, 1, 3, -3, 1, 0, -2, 1, -4, 2, 4, -4, 1, 0, 0, 1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 5, -5, -5, 5, 5, -5, 1, 0, 0, 0, -2, 1, -6, 6, 6, 3, -2, -6, -12, 9, 6, -6, 1, 0, -4, 0, 2, 4, -4, 1, 0, 0, -6, 6, 3, -5, 1, 0, 0, 0, 0, 1, 7, -7, -7, -7, 14, 7, 7

Offset

1,3

Comments

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Table of n, a(n) for n=1..79.

Wikipedia, Symmetric polynomial

Example

Triangle begins:
   1
   1
  -2   1
   0   1
   3  -3   1
   0  -2   1
  -4   2   4  -4   1
   0   0   1
   0   4   0  -4   1
   0   0   3  -3   1
   5  -5  -5   5   5  -5   1
   0   0   0  -2   1
  -6   6   6   3  -2  -6 -12   9   6  -6   1
   0  -4   0   2   4  -4   1
   0   0  -6   6   3  -5   1
   0   0   0   0   1
   7  -7  -7  -7  14   7   7   7  -7  -7 -21  14   7  -7   1
   0   0   0   4   0  -4   1
For example, row 15 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).

Crossrefs

Row sums are A321753.

Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A319225, A319226, A321742-A321765, A321854.

Sequence in context: A321919 A321918 A321754 * A349839 A247919 A127839

Adjacent sequences: A321749 A321750 A321751 * A321753 A321754 A321755

Keyword

sign,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved