A327890 - OEIS

%I #12 Dec 18 2020 04:02:23

%S 0,0,3,6,21,42,90,176,348,640,1203,2152,3848,6692,11701,19968,33966,

%T 56952,95300,157326,258736,421240,683804,1099830,1762867,2805154,

%U 4446826,7005486,10999634,17172894,26716627,41362952,63837722,98079482,150216194,229155682

%N Number of colored integer partitions of n using all colors of a 2-set 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 Vaclav Kotesovec, <a href="/A327890/b327890.txt">Table of n, a(n) for n = 0..4000</a> (terms 0..1000 from Alois P. Heinz)

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

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

%p       `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*

%p        binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))

%p     end:

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

%p seq(a(n), n=0..44);

%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] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];

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

%t a /@ Range[0, 44] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A309973.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 29 2019