cp's OEIS Frontend

A327557 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

%I A327557 #9 Dec 16 2020 16:04:41
%S A327557 0,1,5,29,173,1129,7933,59757,480389,4102233,37059485,352891285,
%T A327557 3530465753,37001007337,405191214949,4625525704837,54929552638957,
%U A327557 677283511701937,8655757492783861,114479050583748677,1564613481125976373,22068492671782019793
%N A327557 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.
%H A327557 Alois P. Heinz, <a href="/A327557/b327557.txt">Table of n, a(n) for n = 0..300</a>
%F A327557 a(n) = Sum_{k=1..n} k * A327116(n,k).
%p A327557 C:= binomial:
%p A327557 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327557       b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
%p A327557     end:
%p A327557 a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
%p A327557 seq(a(n), n=0..23);
%t A327557 c = Binomial;
%t A327557 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
%t A327557 a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}], {k, 0, n}];
%t A327557 a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%Y A327557 Cf. A327116.
%K A327557 nonn
%O A327557 0,3
%A A327557 _Alois P. Heinz_, Sep 16 2019