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A037293 a(n) = Sum_{i=0..2^(n-1)} binomial(2^(n-1), i)^2.

%I A037293 #78 May 02 2025 01:26:36
%S A037293 1,2,6,70,12870,601080390,1832624140942590534,
%T A037293 23951146041928082866135587776380551750,
%U A037293 5768658823449206338089748357862286887740211701975162032608436567264518750790
%N A037293 a(n) = Sum_{i=0..2^(n-1)} binomial(2^(n-1), i)^2.
%C A037293 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 A037293 Seiichi Manyama, <a href="/A037293/b037293.txt">Table of n, a(n) for n = 0..11</a>
%H A037293 Aniruddha Biswas and Palash Sarkar, <a href="https://arxiv.org/abs/2304.14069">Counting unate and balanced monotone Boolean functions</a>, arXiv:2304.14069 [math.CO], 2023.
%H A037293 Aniruddha Biswas and Palash Sarkar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Biswas/biswas6.html">Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 2.
%H A037293 Niklas 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 A037293 a(n) = A001405(2^n). - _Labos Elemer_, Apr 11 2001
%F A037293 a(n) ~ 2^(2^n - n/2 + 1/2)/ sqrt(Pi). - _Vaclav Kotesovec_, Nov 13 2014
%F A037293 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 A037293 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 A037293 a(n) = 2 * A069954(n-1) for n > 0. - _Tilman Piesk_, Oct 11 2024
%p A037293 a:= n-> (t-> binomial(t, iquo(t, 2)))(2^n):
%p A037293 seq(a(n), n=0..8);  # _Alois P. Heinz_, Jan 14 2017
%p A037293 #
%p A037293 a:=n->sum(binomial(2^(n-1),i)^2,i=0..2^(n-1)); seq(a(n),n=0..8);
%p A037293 a:=n->piecewise(n=0,1,binomial(2^n,2^(n-1))); seq(a(n),n=0..8); # _Martin Renner_, Jan 16 2017
%t A037293 Flatten[{1,Table[Binomial[2^n, 2^(n-1)], {n, 1, 8}]}] (* _Vaclav Kotesovec_, Nov 13 2014 *)
%o A037293 (PARI) a(n) = sum(i=0, 2^(n-1), binomial(2^(n-1), i)^2)  \\ _Michel Marcus_, Jun 09 2013
%o A037293 (Python)
%o A037293 from math import comb
%o A037293 def A037293(n): return comb(1<<n,1<<n-1) if n else 1 # _Chai Wah Wu_, Dec 11 2024
%Y A037293 Cf. A000079, A000984, A001405, A069954, A000721.
%K A037293 nonn,easy
%O A037293 0,2
%A A037293 _John Tromp_, Dec 11 1999
%E A037293 More terms from _Erich Friedman_