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A237999 Number of partitions of 2^n into parts that are at most n with at least one part of each size. 6
0, 1, 1, 2, 9, 119, 4935, 596763, 211517867, 224663223092, 734961197081208, 7614278809664610952, 256261752606028225485183, 28642174350851846128820426827, 10830277060032417592098008847162727, 14068379226083299071248895931891435683229 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A316855

Adjacent sequences:  A237996 A237997 A237998 * A238000 A238001 A238002

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 16 2014

STATUS

approved

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Last modified May 28 04:58 EDT 2020. Contains 334671 sequences. (Running on oeis4.)