A321754 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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:

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