A327843 - OEIS

%I #9 Dec 18 2020 04:02:02

%S 1,1,7,94,2081,67390,2969647,169299808,12032189630,1036485156029,

%T 105880393642170,12604896326749405,1724189631362670619,

%U 267831346979691504798,46782781937811822181581,9111872329195713764645644,1964607669245374038857479576

%N Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.

%H Alois P. Heinz, <a href="/A327843/b327843.txt">Table of n, a(n) for n = 0..160</a>

%F a(n) = A327117(2n,n).

%e a(2) = 7: 2ab2ab, 2ab1a1a, 2ab1a1b, 2ab1b1b 1a1a1a1b, 1a1a1b1b, 1a1b1b1b.

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

%p       add(b(n-i*j, min(n-i*j, i-1), k)*binomial(

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

%p     end:

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

%p seq(a(n), n=0..17);

%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] + j - 1, j], {j, 0, n/i}]]];

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

%t a /@ Range[0, 17] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)

%Y Cf. A327117.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 27 2019