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

1, 1, 6, 88, 2489, 112669, 8204101, 799422247, 109633217402, 19157475773052, 4260985739868007, 1161511740640164091, 388990633971649889649, 152369510393132343133762, 70914541309488196549283707, 38152280583855500772704976704, 23639325145221113389859164367779

Offset: 0

Author

Alois P. Heinz, Sep 20 2019

Keywords

• nonn

Comments

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.

Examples

			a(2) = 6: 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..232

Crossrefs

Cf. A327639.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..17);

Mathematica

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]];
a[n_] := Sum[b[2n, 2n, i] (-1)^(n - i) Binomial[n, i], {i, 0, n}];
a /@ Range[0, 17] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) = A327639(2n,n).