%I #11 Jun 24 2019 18:25:17

%S 1,1,-1,-2,0,-2,6,0,-3,6,-9,24,0,0,0,0,0,-24,120,0,0,-40,0,0,-30,80,

%T 90,-90,-130,720,0,0,0,0,0,0,0,0,0,0,0,0,0,-720,5040,0,0,0,-1260,0,0,

%U 0,1260,0,0,2520,3780,0,-945,3780,0,0,0,-6930,6300,-8505

%N Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n, multiplied by n!, in terms of the power sum symmetric functions (using partitions in the Abramowitz-Stegun order)

%C The length of row n is A000041(n).

%C 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.

%C 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).

%C 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.

%C 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).

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=821">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]

%H H J. Borger, <a href="https://arxiv.org/abs/1310.3013">Witt vectors, semirings, and total positivity</a>, 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]

%H SAGE, <a href="https://www.math.sciences.univ-nantes.fr/~sorger/chow/html/en/reference/combinat/sage/combinat/sf/witt.html">Witt symmetric functions</a>

%F 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.

%F 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*an,j) = n. The number of parts is Sum_{j=1..n} a(n,k) =: m(k).

%e The irregular triangle (partition array) begins:

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

%e ---------------------------------------------------------------

%e 1: 1

%e 2: 1 -1

%e 3: -2 0 -2

%e 4: 6 0 -3 6 -9

%e 5: 24 0 0 0 0 0 -24

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

%e 7: 720 0 0 0 0 0 0 0 0 0 0 0 0 0 -720

%e ...

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

%e 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;

%e 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;

%e ...

%e ---------------------------------------------------------------

%e w_1 = p_1;

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

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

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

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

%e 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)

%e = (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);

%e ...

%e ---------------------------------------------------------------

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

%K sign,tabf

%O 1,4

%A _Wolfdieter Lang_, Jun 05 2019