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A326396 Total number of colors in all series-reduced rooted trees with n leaves where colors span an initial interval of the color palette.

%I A326396 #18 Dec 22 2020 03:54:25
%S A326396 1,3,26,322,5210,104421,2491498,68907073,2166242180,76266794945,
%T A326396 2972079029674,126987589678185,5902427979920102,296484317531254557,
%U A326396 16003975713659818226,923838934059255332723,56788871072327503930862,3703444074072753204057172
%N A326396 Total number of colors in all series-reduced rooted trees with n leaves where colors span an initial interval of the color palette.
%H A326396 Alois P. Heinz, <a href="/A326396/b326396.txt">Table of n, a(n) for n = 1..200</a>
%F A326396 a(n) = Sum_{k=1..n} k * A319376(n,k).
%F A326396 From _Vaclav Kotesovec_, Sep 18 2019: (Start)
%F A326396 a(n) ~ c * d^n * n^n, where d = 1.37392076830840090205551979... and c = 0.29889555940946459367729...
%F A326396 a(n) ~ n*A316651(n)/(2*log(2)). (End)
%p A326396 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A326396       add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A326396     end:
%p A326396 A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
%p A326396 a:= n-> add(k*add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1), k=1..n):
%p A326396 seq(a(n), n=1..20);
%t A326396 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
%t A326396     Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
%t A326396 A [n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
%t A326396 a[n_] := Sum[k Sum[A[n, k-j](-1)^j Binomial[k, j], {j, 0, k-1}], {k, 1, n}];
%t A326396 Array[a, 20] (* _Jean-François Alcover_, Dec 22 2020, after _Alois P. Heinz_ *)
%Y A326396 Cf. A316651, A319376.
%K A326396 nonn
%O A326396 1,2
%A A326396 _Alois P. Heinz_, Sep 11 2019