OFFSET
0,3
LINKS
Tom Copeland, Self-Convolution of the Permutohedral Polynomials, 2022.
FORMULA
Rows e.g.f.: (3 - G(x))/2, where G(x) = 1 / (1 + a_1 x + a_2 x^2/2! + a_3 x^3/3! + ...)^2.
d/da_n G(x) = 2 (x^n/n!) (G(x) - 1/2) = 2 (x^n/n!) (-1/2 + P_1 x + P_2 x^2/2! + ...).
EXAMPLE
The first few rows of coefficients, with lengths given by A000041, are
0) 1;
1) 1;
2) -3, 1;
3) 12, -9, 1;
4) -60, 72, -9, -12, 1;
5) 360, -600, 180, 120, -30, -15, 1;
6) -2520, 5400, -2700, -1200, 180, 720, 180, -30, -45, -18, 1;
7) 20160, -52920, 37800, 12600, -6300, -12600, -2100, 1260, 840, 1260, 252, -105, -63, -21, 1;
8) -181440, 564480, -529200, -141120, 151200, 201600, 25200, -6300, -50400, -16800, -25200, -3360, 3360, 2520, 3360, 2016, 336, -105, -168, -84, -24, 1;
... .
The first few partition polynomials with monomials in reverse order to those of Abramowitz and Stegun (p. 831-2, see link in A000041) are
P_0 = 1
P_1(a_1)= 1 a_1
P_2(a_1,a_2) = -3 a_1^2 + 1 a_2
P_3(a_1,a_2,a_3) = 12 a_1^3 - 9 a_1 a_2 + 1 a_3
P_4(a_1,a_2,a_3,a_4) = -60 a_1^4 + 72 a_1^2 a_2 - 9 a_2^2 -12 a_1 a_3 + 1 a_4
P_5 = 360 a_1^5 - 600 a_1^3 a_2 + 180 a_1 a_2^2 + 120 a_1^2 a_3 - 30 a_2 a_3 - 15 a_1 a_4 + 1 a_5
P_6 = -2520 a_1^6 + 5400 a_1^4 a_2 - 2700 a_1^2 a_2^2 - 1200 a_1^3 a_3 + 180 a_2^3 + 720 a_1 a_2 a_3 + 180 a_1^2 a_4 - 30 a_3^2 - 45 a_2 a_4 - 18 a_1 a_5 + 1 a_6
P_7 = 20160 a_1^7 - 52920 a_1^5 a_2 + 37800 a_1^3 a_2^2 12600 a_1^4 a_3 + - 6300 a_1 a_2^3 - 12600 a_1^2 a_2 a_3 - 2100 a_1^3 a_4 + 1260 a_2^2 a_3 + 840 a_1 a_3^2 + 1260 a_1 a_2 a_4 + 252 a_1^2 a_5 - 105 a_3 a_4 - 63 a_2 a_5 - 21 a_1 a_6 + 1 a_7
P_8 = -181440 a_1^8 + 564480 a_1^6 a_2 - 529200 a_1^4 a_2^2 - 141120 a_1^5 a_3 + 151200 a_1^2 a_2^3 + 201600 a_1^3 a_2 a_3 + 25200 a_1^4 a_4 - 6300 a_2^4 - 50400 a_1 a_2^2 a_3 - 16800 a_1^2 a_3^2 - 25200 a_1^2 a_2 a_4 - 3360 a_1^3 a_5 + 3360 a_2 a_3^2 + 2520 a_2^2 a_4 + 3360 a_1 a_3 a_4 + 2016 a_1 a_2 a_5 + 336 a_1^2 a_6 - 105 a_4^2 - 168 a_3 a_5 - 84 a_2 a_6 - 24 a_1 a_7 + 1 a_8.
MATHEMATICA
rows[n_] := {{1}}~Join~With[{s = 1/(1 + Sum[a[k] x^k/k!, {k, n}] + O[x]^(n+1))^2}, -Table[Expand@Coefficient[k! s, x^k Product[a[t], {t, p}]], {k, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k]]}]/2];
rows[7] // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Tom Copeland, Jul 27 2022
EXTENSIONS
Ordering in row 7 changed and formula edited by Andrey Zabolotskiy, Mar 07 2024
STATUS
approved