A327550 - OEIS

%I #23 Apr 24 2021 09:55:41

%S 1,2,8,24,80,224,704,1920,5632,15360,43008,114688,315392,827392,

%T 2211840,5767168,15138816,38928384,100925440,256901120,657457152,

%U 1660944384,4202692608,10527703040,26424115200,65699577856,163477192704,403995361280,998043025408

%N Number of compositions of partitions of 2n with exactly n compositions.

%H Alois P. Heinz, <a href="/A327550/b327550.txt">Table of n, a(n) for n = 0..3129</a>

%F a(n) = A327549(2n,n).

%F a(n) ~ 2^n * exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - _Vaclav Kotesovec_, Sep 19 2019

%F a(n) = A000041(n) * A000079(n). - _Alois P. Heinz_, Oct 08 2020

%F G.f.: Product_{k>=1} 1 / (1 - 2^k*x^k). - _Ilya Gutkovskiy_, Apr 24 2021

%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or

%p       k<1, 0, b(n, i-1, k)+ 2^(i-1)*b(n-i, min(n-i, i), k-1)))

%p     end:

%p a:= n-> b(2*n$2, n)-`if`(n=0, 0, b(2*n$2, n-1)):

%p seq(a(n), n=0..40);

%t Table[2^n * PartitionsP[n], {n, 0, 30}] (* _Vaclav Kotesovec_, Oct 02 2020 *)

%Y Cf. A000041, A000079, A327549.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 16 2019