A321754 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of h(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and h is homogeneous 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

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).

Up to sign, same as A321752.

Links

Table of n, a(n) for n=1..77.

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) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Crossrefs

Row sums are all equal to 1.

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

Sequence in context: A353279 A321919 A321918 * A321752 A349839 A247919

Adjacent sequences: A321751 A321752 A321753 * A321755 A321756 A321757

Keyword

sign,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved