login
A327890
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.
2
0, 0, 3, 6, 21, 42, 90, 176, 348, 640, 1203, 2152, 3848, 6692, 11701, 19968, 33966, 56952, 95300, 157326, 258736, 421240, 683804, 1099830, 1762867, 2805154, 4446826, 7005486, 10999634, 17172894, 26716627, 41362952, 63837722, 98079482, 150216194, 229155682
OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..4000 (terms 0..1000 from Alois P. Heinz)
EXAMPLE
a(3) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
end:
a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=0..44);
MATHEMATICA
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}]]];
a[n_] := With[{k = 2}, Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}]];
a /@ Range[0, 44] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)
CROSSREFS
Column k=2 of A309973.
Sequence in context: A056489 A015649 A297189 * A076102 A094282 A124493
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 29 2019
STATUS
approved