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A297569 Number of nonisomorphic proper colorings of partition star graph using five colors.

%I A297569 #22 Jan 17 2018 13:15:28
%S A297569 5,20,50,80,100,320,320,175,800,680,1280,1280,280,1600,2720,3200,5120,
%T A297569 5120,5120,420,2800,6800,4080,6400,20480,10400,12800,20480,20480,
%U A297569 20480,600,4480,13600,16320,11200,51200,43520,41600,25600,81920,81920,51200,81920,81920,81920,825,6720,23800,40800,19380,17920,102400,174080,104000,166400,44800,204800,174080,327680,164480,102400,327680,327680,204800,327680,327680
%N A297569 Number of nonisomorphic proper colorings of partition star graph using five colors.
%C A297569 A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.
%H A297569 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2584928/">Orbital chromatic polynomials</a>
%F A297569 For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=5.
%e A297569 Rows are:
%e A297569     5;
%e A297569    20,
%e A297569    50,   80;
%e A297569   100,  320,  320;
%e A297569   175,  800,  680, 1280, 1280;
%e A297569   280, 1600, 2720, 3200, 5120, 5120, 5120;
%p A297569 b:= (n, i)-> `if`(n=0, [5], `if`(i<1, [], [seq(map(x-> x*
%p A297569      binomial(4^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
%p A297569 T:= n-> b(n$2)[]:
%p A297569 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jan 14 2018
%t A297569 b[n_, i_] := If[n == 0, {5}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 4^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
%t A297569 T[n_] := b[n, n];
%t A297569 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 17 2018, after _Alois P. Heinz_ *)
%Y A297569 Cf. A297567, A297568, A297570.
%Y A297569 Row sums give 5*A144068.
%Y A297569 Row lengths give A000041.
%K A297569 nonn,tabf
%O A297569 0,1
%A A297569 _Marko Riedel_, Dec 31 2017