A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

1, 1, 2, 10, 55, 412, 3736, 40518, 505486, 7145031, 112844566, 1970286922, 37676184205

Offset

0,3

Links

Table of n, a(n) for n=0..12.

Altug Alkan, Line plot of t(n)-t(n-1) where t(n) = a(n+1)/a(n) for n <= 11

Formula

a(n) >= A272363(n).

Example

For n = 2, a(2) = 2 since {(1,3), (2,4)} and {(1,2), (3,4)} are corresponding sets. {(2,4), (1,3)} and {(3,1), (4,2)} are considered to be the same grouping with {(1,3), (2,4)}.

Prog

(PARI)
okperm(vp, n) = {for (k=1, n-1, if (vp[k] > vp[k+1], return (0)); ); for (k=1, n, if (vp[k+n] <= vp[k], return (0)); ); 1; }
a(n) = if(n==0, 1, {nb = 0; nn = 2*n; for (j=0, nn!-1, vp = numtoperm(nn, j); if (okperm(vp, n), vs = vector(n, k, vp[k]+vp[k+n]); if (#vs == #Set(concat(vs)), nb++); ); ); nb; } ) \\ after Michel Marcus at A272363
(Python)
from sympy.utilities.iterables import multiset_partitions
def A320129(n):
    return 1 if n == 0 else sum(1 for p in multiset_partitions(list(range(1, 2*n+1)), n) if max(len(d) for d in p) == 2 and len(set(sum(d) for d in p)) == n) # Chai Wah Wu, Oct 08 2018

Crossrefs

Cf. A002968, A060963, A272363.

Sequence in context: A175935 A216721 A122826 * A329871 A108490 A323935

Adjacent sequences: A320126 A320127 A320128 * A320130 A320131 A320132

Keyword

nonn,more

Author

Altug Alkan, Oct 06 2018

Extensions

a(11)-a(12) from Giovanni Resta, Oct 09 2018

Status

approved