%I #32 Dec 09 2020 09:01:07

%S 0,1,4,19,98,570,3642,25292,189454,1519648,12978141,117437020,

%T 1121299471,11256640012,118443403699,1302670531063,14938986954323,

%U 178248001223476,2208487163394749,28363722744050886,376991516806826090,5178009641895235269,73396161423153313320

%N Total number of colors used 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 distinct colors in increasing order.

%H Alois P. Heinz, <a href="/A327115/b327115.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = Sum_{k=1..n} A326914(n,k) = Sum_{k=1..n} A326962(n,k).

%e a(2) = 4: 2ab, 1a1b. Both colors (a and b) are used twice: 2 + 2 = 4.

%p C:= binomial:

%p g:= proc(n) option remember; n*2^(n-1) end:

%p h:= proc(n) option remember; local k; for k from

%p `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od

%p end:

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<h(n),

%p 0, add(b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))

%p end:

%p a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=h(n)..n):

%p seq(a(n), n=0..23);

%t c = Binomial;

%t g[n_] := g[n] = n 2^(n - 1);

%t h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]] , True, k++, If [g[k] >= n , Return[k]]]];

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < h[n], 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];

%t a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k-i) c[k, i], {i, 0, k}], {k, h[n], n}];

%t a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)

%Y Cf. A326914, A326962.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 13 2019