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%I #76 Apr 21 2024 09:58:34
%S 0,1,6,27,110,430,1652,6307,24054,91866,351692,1350030,5196204,
%T 20050108,77542376,300507427,1166737574,4537436578,17672369756,
%U 68922740122,269127888644,1052047384708,4116711169496,16123793452942,63205286441660,247959232919620,973469645715192
%N Number of unordered pairs of distinct length-n binary words having the same number of 1's.
%C Equals row sums of triangle A143418, starting with a(2). - _Gary W. Adamson_, Aug 14 2008
%C In coupled systems of n spin 1/2 particles (magnetic resonance) where the spin state of the i-th particle can be coded as 0 (Sz_i=-1/2) or 1 (Sz_i=+1/2), number of distinct (v<w) nontrivial (v!=w) zero-quantum transitions (v->w). - _Stanislav Sykora_, Jun 07 2012
%C a(n) is the number of lattice paths from (0,0) to (n,n) using E(1,0) and N(0,1) as steps that horizontally cross the diagonal y = x with odd many times. For example, a(2) = 1 because there is only one path that horizontally crosses the diagonal with odd many times, namely, NEEN. - _Ran Pan_, Feb 01 2016
%H Michael De Vlieger, <a href="/A108958/b108958.txt">Table of n, a(n) for n = 1..1664</a>
%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca2/merca7.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
%H Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%H Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
%F a(n) = Sum_{k=0..n} binomial(binomial(n, k), 2).
%F From _Vladeta Jovovic_, Jul 24 2005: (Start)
%F a(n) = binomial(2*n-1, n-1)-2^(n-1) = A088218(n) - A011782(n).
%F E.g.f.: exp(2*x)*(BesselI(0, 2*x)-1)/2. (End)
%F a(n) = (1/2)*Sum_{i+j>n,0<=i,j<=n} binomial(i+j,i). - _Benoit Cloitre_, May 26 2006
%F Conjecture: n*(n-2)*a(n) +2*(-3*n^2+7*n-3)*a(n-1) +4*(n-1)*(2*n-3) *a(n-2)=0. - _R. J. Mathar_, Apr 04 2012
%F a(n) = Sum_{0<i<=k<n} (-1)^(i+1)*binomial(n,k+i)*binomial(n,k-i). - _Mircea Merca_, Apr 05 2012
%F a(n) = binomial(2*n,n) - A005317(n), - _Ran Pan_, Feb 01 2016
%F a(n) = 1/2*Sum_{k=1..n} binomial(n,k)^2 - binomial(n,k). - _Gerry Martens_, Oct 09 2022
%F a(n) ~ 2^(2*n-1)/sqrt(n*Pi). - _Stefano Spezia_, Apr 17 2024
%e a(3) = 6 because the pairs are {001,010}, {001,100}, {010,100}, {011,101}, {011,110}, {101,110}.
%p with(combinat) a:= proc(n) add(binomial(binomial(n,k), 2), k=0..n) end;
%t Table[Binomial[2 n, n] - (2^n + Binomial[2 n, n])/2, {n, 30}] (* _Vincenzo Librandi_, Feb 01 2016 *)
%o (Magma) [Binomial(2*n,n)-(2^n+Binomial(2*n,n))/2: n in [1..30]]; // _Vincenzo Librandi_, Feb 01 2016
%o (PARI) a(n)=binomial(2*n-1,n-1)-2^(n-1) \\ _Charles R Greathouse IV_, Feb 01 2016
%o (Python)
%o from math import comb
%o def A108958(n): return comb((n<<1)-1,n-1)-(1<<n-1) # _Chai Wah Wu_, Sep 23 2022
%Y Cf. A005317, A011782, A088218, A143418.
%K easy,nonn,changed
%O 1,3
%A _Jeffrey Shallit_, Jul 22 2005
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