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 A324253 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) 1
 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, 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, 0, 1260, 0, 0, 2520, 3780, 0, -945, 3780, 0, 0, 0, -6930, 6300, -8505 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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). LINKS 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 FORMULA 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*an,j) = n. The number of parts is Sum_{j=1..n} a(n,k) =: m(k). EXAMPLE 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); ... --------------------------------------------------------------- CROSSREFS Cf. A000041, A115131 (Waring numbers), A324247. Sequence in context: A264954 A212085 A265882 * A208385 A186634 A139213 Adjacent sequences:  A324250 A324251 A324252 * A324254 A324255 A324256 KEYWORD sign,tabf AUTHOR Wolfdieter Lang, Jun 05 2019 STATUS approved

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Last modified May 21 05:37 EDT 2022. Contains 353889 sequences. (Running on oeis4.)