A261732 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(list; graph; refs; listen; history; text; internal format)

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.