A237999 - OEIS

%I #22 May 31 2019 20:13:51

%S 0,1,1,2,9,119,4935,596763,211517867,224663223092,734961197081208,

%T 7614278809664610952,256261752606028225485183,

%U 28642174350851846128820426827,10830277060032417592098008847162727,14068379226083299071248895931891435683229

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

%C From _Gus Wiseman_, May 31 2019: (Start)

%C 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):

%C   (2)  (31)  (431)  (6532)

%C              (521)  (6541)

%C                     (7432)

%C                     (7531)

%C                     (7621)

%C                     (8431)

%C                     (8521)

%C                     (9421)

%C                     (A321)

%C (End)

%H Alois P. Heinz, <a href="/A237999/b237999.txt">Table of n, a(n) for n = 0..62</a>

%H A. V. Sills and D. Zeilberger, <a href="https://arxiv.org/abs/1108.4391">Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)</a>, arXiv:1108.4391 [math.CO], 2011.

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

%F a(n) ~ 2^(n*(n-1)) / (n!*(n-1)!). - _Vaclav Kotesovec_, Jun 05 2015

%e a(1) = 1: 11.

%e a(2) = 1: 211.

%e a(3) = 2: 3221, 32111.

%e a(4) = 9: 433321, 443221, 4322221, 4332211, 4432111, 43222111, 43321111, 432211111, 4321111111.

%t a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, 2^n - n*(n + 1)/2}];

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* _Jean-François Alcover_, Aug 19 2018 *)

%Y Column k=2 of A238012.

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

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

%K nonn

%O 0,4

%A _Alois P. Heinz_, Feb 16 2014