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%I #32 Sep 08 2022 08:45:01
%S 0,4,80,1408,23808,395264,6496256,106102784,1725759488,27987017728,
%T 452890460160,7316694892544,118053383700480,1902813588226048,
%U 30644438111879168,493182667873845248,7932561728196313088,127528427796392050688,2049363966872621416448
%N a(n) = 2^(4*n-1) - 2^(2*n-1)*binomial(2*n,n).
%C Number of 4-ary words of length 2n in which the combined count of 0's and 1's is greater than n. - _David Scambler_, Aug 13 2012
%D The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1982, (3.98), page 33.
%H G. C. Greubel, <a href="/A055787/b055787.txt">Table of n, a(n) for n = 0..825</a>
%H H. W. Gould, ed. J. Quaintance, <a href="http://www.math.wvu.edu/~gould/Vol.4.PDF">Combinatorial Identities</a>, May 2010, (7.6) p. 35.
%F a(n) + A045952(n) = 2^(4n). - _David Scambler_, Aug 13 2012
%F a(n) = Sum_{k=0..n-1} binomial(4*k+2, 2*k+1) * binomial(4*n-4*k-2, 2*n-2*k-1). (LHS of Gould's identity, corrected).
%F a(n) = 2^(4*n-1)*(1-(Gamma(n+1/2))/(sqrt(Pi)*Gamma(n+1))). - _Alexander R. Povolotsky_, Aug 13 2012
%F a(n) = Sum_{k=0..n-1}(2^(2*n) * C(2*n, k)). - _David Scambler_, Aug 13 2012
%F a(n+2) = (8*(4*n+5)*a(n+1))/(n+2) - (128*(2*n+1)*a(n))/(n+2). - _Alexander R. Povolotsky_, Aug 14 2012
%F G.f.: (1 - sqrt(1-16*x))/(2*(1-16*x)). - _G. C. Greubel_, Jan 25 2020
%p seq( 2^(4*n-1) - 2^(2*n-1)*binomial(2*n,n), n=0..20); # _G. C. Greubel_, Jan 25 2020
%t Table[2^(4*n-1) - 2^(2*n-1)*Binomial[2*n, n], {n,0,20}] (* _G. C. Greubel_, Feb 16 2017 *)
%o (PARI) vector(21, n, my(m=n-1); 2^(4*m-1) -2^(2*m-1)*binomial(2*m, m) ) \\ _G. C. Greubel_, Feb 16 2017
%o (Magma) [2^(4*n-1) - 2^(2*n-1)*Binomial(2*n,n): n in [0..20]]; // _G. C. Greubel_, Jan 25 2020
%o (Sage) [2^(4*n-1) - 2^(2*n-1)*binomial(2*n,n) for n in (0..20)] # _G. C. Greubel_, Jan 25 2020
%o (GAP) List([0..20], n-> 2^(4*n-1) - 2^(2*n-1)*Binomial(2*n,n)); # _G. C. Greubel_, Jan 25 2020
%Y Cf. A045952.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jul 15 2000
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