A327598 - OEIS

%I #32 Dec 17 2020 07:55:18

%S 0,0,2,6,15,32,65,124,230,414,729,1258,2141,3586,5935,9716,15738,

%T 25258,40196,63452,99426,154732,239219,367592,561602,853300,1289777,

%U 1939920,2904003,4327672,6421572,9489260,13967003,20479638,29919253,43556102,63193528

%N 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 Alois P. Heinz, <a href="/A327598/b327598.txt">Table of n, a(n) for n = 0..3000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%e a(2) = 2: 2ab, 1a1b.

%e a(3) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.

%p C:= binomial:

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

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

%p     end:

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

%p seq(a(n), n=0..37);

%t c = Binomial;

%t 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 a[n_] := With[{k = 2}, Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}]];

%t a /@ Range[0, 37] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A327116.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 27 2019