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A327598 Number of colored integer partitions of n using all colors of a 2-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

%I A327598 #32 Dec 17 2020 07:55:18
%S A327598 0,0,2,6,15,32,65,124,230,414,729,1258,2141,3586,5935,9716,15738,
%T A327598 25258,40196,63452,99426,154732,239219,367592,561602,853300,1289777,
%U A327598 1939920,2904003,4327672,6421572,9489260,13967003,20479638,29919253,43556102,63193528
%N A327598 Number of colored integer partitions of n using all colors of a 2-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.
%H A327598 Alois P. Heinz, <a href="/A327598/b327598.txt">Table of n, a(n) for n = 0..3000</a>
%H A327598 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A327598 a(2) = 2: 2ab, 1a1b.
%e A327598 a(3) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
%p A327598 C:= binomial:
%p A327598 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327598       b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
%p A327598     end:
%p A327598 a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k))(2):
%p A327598 seq(a(n), n=0..37);
%t A327598 c = Binomial;
%t A327598 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 A327598 a[n_] := With[{k = 2}, Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}]];
%t A327598 a /@ Range[0, 37] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A327598 Column k=2 of A327116.
%K A327598 nonn
%O A327598 0,3
%A A327598 _Alois P. Heinz_, Sep 27 2019