A261732 Number of partitions of 2n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur at least once in the partition.

1, 2, 40, 1751, 136516, 16993932, 3112631737, 792794568624, 269047552566188, 117558189248146187, 64343348623810658670, 43136101281780352785381, 34769493663713954019980281, 33175795620329111188048543481, 36981665738825428991431755534146

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) = A261719(2n,n).

a(n) ~ c * d^n * n! * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 6.1765546094834803582316801640508765536... and c = 0.52251062602313321387485... . - Vaclav Kotesovec, Feb 18 2017

Example

a(0) = 1: the empty partition.
a(1) = 2: 2aa, 1a1a.
a(2) = 40: 4aaab, 4aabb, 4abbb, 3aaa1b, 3aab1a, 3aab1b, 3abb1a, 3abb1b, 3bbb1a, 2aa2ab, 2aa2bb, 2ab2aa, 2ab2ab, 2ab2bb, 2bb2aa, 2bb2ab, 2aa1a1b, 2aa1b1a, 2aa1b1b, 2ab1a1a, 2ab1a1b, 2ab1b1a, 2ab1b1b, 2bb1a1a, 2bb1a1b, 2bb1b1a, 1a1a1a1b, 1a1a1b1a, 1a1a1b1b, 1a1b1a1a, 1a1b1a1b, 1a1b1b1a, 1a1b1b1b, 1b1a1a1a, 1b1a1a1b, 1b1a1b1a, 1b1a1b1b, 1b1b1a1a, 1b1b1a1b, 1b1b1b1a.

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..15);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Crossrefs

Cf. A261719, A226775.

Sequence in context: A275931 A099707 A198248 * A292418 A163826 A000816

Adjacent sequences: A261729 A261730 A261731 * A261733 A261734 A261735

Keyword

nonn

Author

Alois P. Heinz, Aug 30 2015

Status

approved