OFFSET
3,1
COMMENTS
See the formulas dated Sep 20 2016 at A133437 for a discussion of these convolution polynomials.
EXAMPLE
The first few partition polynomials are
P(1) = 0
P(2) = 0
P(3,u2) = 2 (2')^2
P(4,u2,u3) = -10 (2')^3 + 5 (2')(3')
P(5,u2,u3,u4) = 42 (2')^4 - 42 (1') (2')^2 (3') + 3 (3')^2 + 6 (2') (4')
P(6,u2,...,u5) = -168 (2')^5 + 252 (2')^3 (3') - 56 (2') (3')^2 - 56 (2')^2 (4') + 7 (3')(4') + 7 (2')(5')
P(7,u2,...,u6) = 660 (2')^6 - 1320 (2')^4 (3') + 540 (2')^2 (3')^2 + 360 (2')^3 (4') - (24 (3')^3 + 144 (2') (3') (4') + 72 (2')^2 (5')) + 4 (4')^2 + 8 (3') (5') + 8 (2') (6')
P(8,u2,...,u7) = -2574 (2')^7 + 6435 (2')^5 (3') - (3960 (2')^3 (3')^2 + 1980 (2')^4 (4')) + 495 (2') (3')^3 + 1485 (2')^2 (3') (4') + 495 (2')^3 (5') - (90 (3')^2 (4') + 90 (2') (4')^2 + 180 (2')(3')(5') + 90 (2')^2 (6')) + 9 (4')(5') + 9 (3')(6') + 9 (2')(7')
...
MATHEMATICA
rows[nn_] := With[{s = InverseSeries[t (1 + Sum[u[k] t^k, {k, nn}] + O[t]^(nn+1))]}, Table[(Length[p]-1) Coefficient[s, t^(n+1) Product[u[w], {w, p}]], {n, nn}, {p, Most@Reverse@Sort[Sort /@ IntegerPartitions[n]]}]];
rows[7] // Flatten (* Andrey Zabolotskiy, Mar 08 2024 *)
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Tom Copeland, Sep 21 2016
EXTENSIONS
Corrected and extended by Andrey Zabolotskiy, Mar 08 2024
STATUS
approved