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A327645 Number of proper n-times partitions of 2n.

%I A327645 #9 Dec 18 2020 04:01:04
%S A327645 1,1,6,88,2489,112669,8204101,799422247,109633217402,19157475773052,
%T A327645 4260985739868007,1161511740640164091,388990633971649889649,
%U A327645 152369510393132343133762,70914541309488196549283707,38152280583855500772704976704,23639325145221113389859164367779
%N A327645 Number of proper n-times partitions of 2n.
%C A327645 In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
%H A327645 Alois P. Heinz, <a href="/A327645/b327645.txt">Table of n, a(n) for n = 0..232</a>
%F A327645 a(n) = A327639(2n,n).
%e A327645 a(2) = 6: 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111.
%p A327645 b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
%p A327645       b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p A327645     end:
%p A327645 a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
%p A327645 seq(a(n), n=0..17);
%t A327645 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
%t A327645 a[n_] := Sum[b[2n, 2n, i] (-1)^(n - i) Binomial[n, i], {i, 0, n}];
%t A327645 a /@ Range[0, 17] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327645 Cf. A327639.
%K A327645 nonn
%O A327645 0,3
%A A327645 _Alois P. Heinz_, Sep 20 2019