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A327679 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

%I A327679 #12 Dec 18 2020 04:01:25
%S A327679 1,1,4,18,112,732,6156,53720,559584,6138216,76636080,1006039320,
%T A327679 14693223032,224774090592,3756082129296,65650522695344,
%U A327679 1236568354232176,24299076684879264,509677108276779168,11124779898457678240,257204596479739401760,6174928911548312072704
%N A327679 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.
%H A327679 Alois P. Heinz, <a href="/A327679/b327679.txt">Table of n, a(n) for n = 0..200</a>
%p A327679 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A327679       `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
%p A327679        binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
%p A327679     end:
%p A327679 a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
%p A327679 seq(a(n), n=0..22);
%t A327679 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]Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
%t A327679 a[n_] := Sum[Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A327679 a /@ Range[0, 22] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327679 Row sums of A309973.
%K A327679 nonn
%O A327679 0,3
%A A327679 _Alois P. Heinz_, Sep 21 2019