A237999 Number of partitions of 2^n into parts that are at most n with at least one part of each size.

0, 1, 1, 2, 9, 119, 4935, 596763, 211517867, 224663223092, 734961197081208, 7614278809664610952, 256261752606028225485183, 28642174350851846128820426827, 10830277060032417592098008847162727, 14068379226083299071248895931891435683229

Offset

0,4

Comments

From Gus Wiseman, May 31 2019: (Start)

Also the number of strict integer partitions of 2^n with n parts. For example, the a(1) = 1 through a(4) = 9 partitions are (A = 10):

(2) (31) (431) (6532)

(521) (6541)

(7432)

(7531)

(7621)

(8431)

(8521)

(9421)

(A321)

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..62

A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.

Formula

a(n) = [x^(2^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 2^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

Example

a(1) = 1: 11.
a(2) = 1: 211.
a(3) = 2: 3221, 32111.
a(4) = 9: 433321, 443221, 4322221, 4332211, 4432111, 43222111, 43321111, 432211111, 4321111111.

Mathematica

a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, 2^n - n*(n + 1)/2}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* Jean-François Alcover, Aug 19 2018 *)

Crossrefs

Column k=2 of A238012.

Cf. A236810, A237512, A237998, A238000, A238001.

Cf. A000009, A002033, A067735, A126796, A283111.

Sequence in context: A201381 A075538 A067965 * A194017 A135543 A372745

Adjacent sequences: A237996 A237997 A237998 * A238000 A238001 A238002

Keyword

nonn

Author

Alois P. Heinz, Feb 16 2014

Status

approved