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%I #40 Mar 05 2022 03:54:53
%S 1,1,3,21,115,813,7627,71173,740023,8544169,107195083,1434581205,
%T 20499413667,312262663989,4992164670007,84221279919193,
%U 1492818584618099,27607818180267269,533522844488072987,10724970103003953053,223859943086157531063,4847766598150865273721
%N Number of colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.
%H Alois P. Heinz, <a href="/A309670/b309670.txt">Table of n, a(n) for n = 0..200</a>
%p C:= binomial:
%p b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
%p b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
%p end:
%p a:= n-> add(add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
%p seq(a(n), n=0..23);
%t c = Binomial;
%t b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]];
%t a[n_] := Sum[Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}];
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Mar 05 2022, after _Alois P. Heinz_ *)
%Y Row sums of A327244.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 18 2019
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