A321896 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in e(u) * Product_i u_i!, where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

1, 1, -1, 1, 0, 1, 2, -3, 1, 0, -1, 1, -6, 3, 8, -6, 1, 0, 0, 1, 0, 1, 0, -2, 1, 0, 0, 2, -3, 1, 24, -30, -20, 15, 20, -10, 1, 0, 0, 0, -1, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 0, -6, 0, 3, 8, -6, 1, 0, 0, -2, 3, 2, -4, 1, 0, 0, 0, 0, 1, 720

Offset

1,7

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..73.

Wikipedia, Symmetric polynomial

Example

Triangle begins:
     1
     1
    -1    1
     0    1
     2   -3    1
     0   -1    1
    -6    3    8   -6    1
     0    0    1
     0    1    0   -2    1
     0    0    2   -3    1
    24  -30  -20   15   20  -10    1
     0    0    0   -1    1
  -120   90  144   40  -15  -90 -120   45   40  -15    1
     0   -6    0    3    8   -6    1
     0    0   -2    3    2   -4    1
     0    0    0    0    1
   720 -840 -504 -420  630  504  210  280 -105 -210 -420  105   70  -21    1
     0    0    0    1    0   -2    1
For example, row 15 gives: 12e(32) = -2p(32) + 3p(221) + 2p(311) - 4p(2111) + p(11111).

Crossrefs

Row sums are A036987.

Cf. A005651, A008480, A056239, A124794, A124795, A135278, A319193, A319225, A319226, A321742-A321765, A321897.

Sequence in context: A030373 A079343 A004566 * A321897 A050074 A346688

Adjacent sequences: A321893 A321894 A321895 * A321897 A321898 A321899

Keyword

sign,tabf

Author

Gus Wiseman, Nov 20 2018

Status

approved