A356146 Coefficients of the partition polynomials that are binomial convolutions of the partition polynomials of A133314, the refined Euler characteristic polynomials of the permutahedra and coefficient polynomials of reciprocals of Taylor series or e.g.f.s. Irregular triangle read by rows with length given by A000041.

1, 1, -3, 1, 12, -9, 1, -60, 72, -9, -12, 1, 360, -600, 180, 120, -30, -15, 1, -2520, 5400, -2700, -1200, 180, 720, 180, -30, -45, -18, 1, 20160, -52920, 37800, 12600, -6300, -12600, -2100, 1260, 840, 1260, 252, -105, -63, -21, 1

Offset: 0

Author

Tom Copeland, Jul 27 2022

Keywords

• sign
• tabf

Examples

			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.
		

Links

• Tom Copeland, Self-Convolution of the Permutohedral Polynomials, 2022.

Crossrefs

Cf. A000041, A133314.

Programs

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 *)

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! + ...).

Extensions

Ordering in row 7 changed and formula edited by Andrey Zabolotskiy, Mar 07 2024