cp's OEIS Frontend

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

%I A276850 #21 Mar 09 2024 11:01:15
%S A276850 2,-10,5,42,-42,3,6,-168,252,-56,-56,7,7,660,-1320,540,360,-24,-144,
%T A276850 -72,4,8,8,-2574,6435,-3960,-1980,495,1485,495,-90,-90,-180,-90,9,9,9,
%U A276850 10010,-30030,25025,10010,-5720,-11440,-2860,165,1980,990,1980,660,-110,-110,-220,-220,-110,5,10,10,10
%N A276850 Convolution of partition polynomials of A133437 related to solutions of the Burgers-Hopf equation.
%C A276850 See the formulas dated Sep 20 2016 at A133437 for a discussion of these convolution polynomials.
%e A276850 The first few partition polynomials are
%e A276850 P(1) = 0
%e A276850 P(2) = 0
%e A276850 P(3,u2) = 2 (2')^2
%e A276850 P(4,u2,u3) = -10 (2')^3 + 5 (2')(3')
%e A276850 P(5,u2,u3,u4) = 42 (2')^4 - 42 (1') (2')^2 (3') + 3 (3')^2 + 6 (2') (4')
%e A276850 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')
%e A276850 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')
%e A276850 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')
%e A276850 ...
%t A276850 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]]}]];
%t A276850 rows[7] // Flatten (* _Andrey Zabolotskiy_, Mar 08 2024 *)
%Y A276850 Cf. A133437.
%K A276850 sign,tabf
%O A276850 3,1
%A A276850 _Tom Copeland_, Sep 21 2016
%E A276850 Corrected and extended by _Andrey Zabolotskiy_, Mar 08 2024