A276850 Convolution of partition polynomials of A133437 related to solutions of the Burgers-Hopf equation.

2, -10, 5, 42, -42, 3, 6, -168, 252, -56, -56, 7, 7, 660, -1320, 540, 360, -24, -144, -72, 4, 8, 8, -2574, 6435, -3960, -1980, 495, 1485, 495, -90, -90, -180, -90, 9, 9, 9, 10010, -30030, 25025, 10010, -5720, -11440, -2860, 165, 1980, 990, 1980, 660, -110, -110, -220, -220, -110, 5, 10, 10, 10

Offset

3,1

Comments

See the formulas dated Sep 20 2016 at A133437 for a discussion of these convolution polynomials.

Links

Table of n, a(n) for n=3..60.

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

Cf. A133437.

Sequence in context: A114232 A070730 A082192 * A033468 A047816 A095845

Adjacent sequences: A276847 A276848 A276849 * A276851 A276852 A276853

Keyword

sign,tabf

Author

Tom Copeland, Sep 21 2016

Extensions

Corrected and extended by Andrey Zabolotskiy, Mar 08 2024

Status

approved