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%I #18 Sep 08 2022 08:45:15
%S 1,2,3,4,7,13,23,44,83,159,306,590,1144,2220,4317,8408,16399,32023,
%T 62601,122498,239924,470304,922612,1811217,3558035,6993883,13755529,
%U 27068914,53294747,104979547,206880514,407866454,804432711,1587177283
%N a(n) = ceiling( 1/2 + (Sum_{i=0..n-1}C(n,i)*C(n,i+1))/2^(n+1) ).
%H G. C. Greubel, <a href="/A099779/b099779.txt">Table of n, a(n) for n = 2..1000</a>
%H Radoslav Derka, Vladimir Buzek, Artur Ekert, <a href="https://cds.cern.ch/record/330025">Universal Algorithm for Optimal Estimation of Quantum States from Finite Ensembles</a>, Phys. Rev. Lett. 80 (1998) 1571-1575.
%F a(n) = ceiling(1/2 + n*binomial(2*n,n) / ((n+1) * 2^(n+1))). - _Vaclav Kotesovec_, Sep 04 2019
%p a:=n->ceil((1/2)+(1/(2^(n+1))*sum(binomial(n,i)*binomial(n,i+1), i=0..n-1))): seq(a(n),n=2..36); # _Emeric Deutsch_, Feb 16 2005
%t Table[Ceiling[1/2 + Sum[Binomial[n,j]*Binomial[n,j+1], {j,0,n-1}]/2^(n+1) ], {n,2,40}] (* _G. C. Greubel_, Sep 04 2019 *)
%t Table[Ceiling[1/2 + n*Binomial[2*n, n] / ((n + 1)*2^(n + 1))], {n, 2, 40}] (* _Vaclav Kotesovec_, Sep 04 2019 *)
%o (PARI) a(n) = ceil(1/2 + sum(j=0,n-1, binomial(n,j)*binomial(n,j+1) )/2^(n+1)); \\ _G. C. Greubel_, Sep 04 2019
%o (Magma) [Ceiling(1/2 + &+[Binomial(n,j)*Binomial(n,j+1): j in [0..n-1] ]/2^(n+1) ): n in [2..40]]; // _G. C. Greubel_, Sep 04 2019
%o (Sage) [ceil(1/2 + sum(binomial(n,j)*binomial(n,j+1) for j in (0..n-1) )/2^(n+1)) for n in (2..40)] # _G. C. Greubel_, Sep 04 2019
%o (GAP) Concatenation([1], List([3..40], n-> Int(3/2 + Sum([0..n], j-> (n-j)*Binomial(n,j)^2/(j+1))/2^(n+1) ))); # _G. C. Greubel_, Sep 04 2019
%K nonn,easy
%O 2,2
%A Sibasish Ghosh, _Simone Severini_, Nov 12 2004
%E More terms from _Emeric Deutsch_, Feb 16 2005
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