login
a(n) = Sum_{i=0..2^(n-1)} binomial(2^(n-1), i)^2.
10

%I #70 Nov 12 2024 20:17:53

%S 1,2,6,70,12870,601080390,1832624140942590534,

%T 23951146041928082866135587776380551750,

%U 5768658823449206338089748357862286887740211701975162032608436567264518750790

%N a(n) = Sum_{i=0..2^(n-1)} binomial(2^(n-1), i)^2.

%C For n >= 1, a(n) is the number of n-variable balanced Boolean functions. A Boolean function is balanced if it takes the values 0 and 1 an equal number of times. - _Aniruddha Biswas_, Nov 05 2024

%H Seiichi Manyama, <a href="/A037293/b037293.txt">Table of n, a(n) for n = 0..11</a>

%H A. Biswas and P. Sarkar, <a href="https://arxiv.org/abs/2304.14069">Counting unate and balanced monotone Boolean functions</a>, arXiv:2304.14069 [math.CO], 2023.

%H N. G. Johansson, <a href="http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A840938">Efficient Simulation of the Deutsch-Jozsa Algorithm</a>, Master's Project, Department of Electrical Engineering & Department of Physics, Chemistry and Biology, Linkoping University, April, 2015. See Eq. (3.15).

%F a(n) = A001405(2^n). - _Labos Elemer_, Apr 11 2001

%F a(n) ~ 2^(2^n - n/2 + 1/2)/ sqrt(Pi). - _Vaclav Kotesovec_, Nov 13 2014

%F a(n) = A000984(2^(n-1)) = binomial(2^n,2^(n-1)) = (2^n)!/((2^(n-1))!)^2 for n > 0. - _Martin Renner_, Jan 16 2017

%F a(n) = (2^(2^n)*(2^n + 2)*(1/2*(2^n + 1))!)/(sqrt(Pi)*(2^n + 1)*(1/2*(2^n + 2))!) = (2^(2^n)*(2^n + 2)*Gamma((2^n+3)/2))/(sqrt(Pi)*(2^n + 1)*Gamma(2^(n-1)+2)) for n > 0. - _Alexander R. Povolotsky_, Nov 19 2022

%F a(n) = 2 * A069954(n-1) for n > 0. - _Tilman Piesk_, Oct 11 2024

%p a:= n-> (t-> binomial(t, iquo(t, 2)))(2^n):

%p seq(a(n), n=0..8); # _Alois P. Heinz_, Jan 14 2017

%p #

%p a:=n->sum(binomial(2^(n-1),i)^2,i=0..2^(n-1)); seq(a(n),n=0..8);

%p a:=n->piecewise(n=0,1,binomial(2^n,2^(n-1))); seq(a(n),n=0..8); # _Martin Renner_, Jan 16 2017

%t Flatten[{1,Table[Binomial[2^n, 2^(n-1)], {n, 1, 8}]}] (* _Vaclav Kotesovec_, Nov 13 2014 *)

%o (PARI) a(n) = sum(i=0, 2^(n-1), binomial(2^(n-1), i)^2) \\ _Michel Marcus_, Jun 09 2013

%Y Cf. A000079, A000984, A001405, A069954, A000721.

%K nonn,easy,changed

%O 0,2

%A _John Tromp_, Dec 11 1999

%E More terms from _Erich Friedman_