1,4

The length of row n is A000041(n).

The (one part) Witt symmetric function w_n is defined in the links below. One should add w_0 = 1. It can be expressed in terms of the power sum symmetric functions p_k = Sum_{i>=1} (x_i)^k for the indeterminates {x_i}, by using the recurrence w_n = (1/n)*(p_n - Sum_{d|n,1 <= d < n} d*(w_d)^{n/d}), n >= 2, with w_1 = p_1.

In order to have integer coefficients n!*w_n is considered, and terms are listed in the Abramowitz-Stegun order (with rising number of parts).

A logarithmic generating function of the power sums is related to the {w_n}_{n>=1} sequence by Lp(t) := -Sum_{j>=1} p_j*(t^j)/j = log(Product_{n>=0} (1 - w_n*t^n)). See the links.

If only N indeterminates {x_i}_{i=1..N} are considered all coefficients corresponding to partitions with at least one part > N are set to 0 (in addition to the ones given in the sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]

H J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015, Section 4.5., pp. 295-296 [with theta -> w, psi-> p, and the n = 1..6 results on p. 295]

SAGE, Witt symmetric functions

w_n is given by the recurrence given in the comment above in terms of the power sum symmetric functions {p_i}_{i>=1}, for n >= 1.

T(n, k) gives the coefficient of (p_1)^{a(1,k)}*...*(p_n)^{a(n,k)} for n!*w_n, corresponding to the k-th partition of n in Abramowitz-Stegun order, written as 1^{a(1,k)}*...*n^{a(n,k)}, with nonnegative integers a(n,j) satisfying Sum_{j=1..n} j*a(n,j) = n. The number of parts is Sum_{j=1..n} a(n,k) =: m(k).

The irregular triangle (partition array) begins:

n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

---------------------------------------------------------------

1: 1

2: 1 -1

3: -2 0 -2

4: 6 0 -3 6 -9

5: 24 0 0 0 0 0 -24

6: 120 0 0 -40 0 0 -30 80 90 -90 -130

7: 720 0 0 0 0 0 0 0 0 0 0 0 0 0 -720

...

n = 8: 5040 0 0 0 -1260 0 0 0 1260 0 0 2520 3780 0 -945 3780 0 0 0 -6930 6300 -8505;

n = 9: 40320 0 0 0 0 0 0 0 0 0 0 -4480 0 0 0 0 0 0 0 0 13440 0 0 0 0 0 -13440 0 0 -35840;

n = 10: 362880 0 0 0 0 -725760 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -22680 145152 0 0 0 113400 0 0 -226800 0 226800 -113400 -412776;

...

---------------------------------------------------------------

w_1 = p_1;

w_2 = (1/2)*(p_2 - (p_1)^2);

w_3 = (1/3!)*(2*p_3 + 0 - 2*(p_1)^3);

w_4 = (1/4!)*(6*p_4 + 0 - 3*(p_2)^2 + 6*(p_1)^2*p_2 - 9*(p_1)^4);

w_5 = (1/5!)*(24*p_5 + 0 + 0 + 0 + 0 + 0 - 24*(p_1)^5) = (1/5)*(p_5 - (p_1)^5);

w_6 = (1/6!)*(120*p_6 + 0 + 0 - 40*(p_3)^2 + 0 + 0 - 30*(p_2)^3 + 80*(p_1)^3*p_3 + 90*(p_1)^2*(p_2)^2 - 90*(p_1)^4*p_2 - 130*(p_1)^6)

= (1/72)*(12*p_6 - 4*(p_3)^2 - 3*(p_2)^3 + 8*(p_1)^3*p_3 + 9*(p_1)^2*(p_2)^2 - 9*(p_1)^4*p_2 - 13*(p_1)^6);

...

---------------------------------------------------------------

Cf. A000041, A115131 (Waring numbers), A324247.

Sequence in context: A212085 A380977 A265882 * A208385 A186634 A363910

Adjacent sequences: A324250 A324251 A324252 * A324254 A324255 A324256

sign,tabf

Wolfdieter Lang, Jun 05 2019

approved