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A108958
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Number of unordered pairs of distinct length-n binary words having the same number of 1's.
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3
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0, 1, 6, 27, 110, 430, 1652, 6307, 24054, 91866, 351692, 1350030, 5196204, 20050108, 77542376, 300507427, 1166737574, 4537436578, 17672369756, 68922740122, 269127888644, 1052047384708, 4116711169496, 16123793452942, 63205286441660, 247959232919620
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OFFSET
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1,3
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COMMENTS
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Equals row sums of triangle A143418, starting with a(2). - Gary W. Adamson, Aug 14 2008
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
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
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LINKS
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Michael De Vlieger, Table of n, a(n) for n = 1..1664
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
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FORMULA
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a(n) = sum(binomial(binomial(n, k), 2), k=0..n);
a(n) = binomial(2*n-1, n-1)-2^(n-1) = A088218(n)-A011782(n). E.g.f.: exp(2*x)*(BesselI(0, 2*x)-1)/2. - Vladeta Jovovic, Jul 24 2005
a(n) = (1/2)*sum(i+j>n,0<=i,j<=n,binomial(i+j,i)). - Benoit Cloitre, May 26 2006
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
a(n) = sum_{0<i<=k<n} (-1)^(i+1)*binomial(n,k+i)*binomial(n,k-i). - Mircea Merca, Apr 05 2012
a(n) = binomial(2*n,n) - A005317(n), - Ran Pan, Feb 01 2016
a(n) = 1/2*Sum_{k=1..n} binomial(n,k)^2 - binomial(n,k). - Gerry Martens, Oct 09 2022
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EXAMPLE
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a(3) = 6 because the pairs are {001,010}, {001,100}, {010,100}, {011,101}, {011,110}, {101,110}.
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MAPLE
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with(combinat) a:= proc(n) add(binomial(binomial(n, k), 2), k=0..n) end;
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MATHEMATICA
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Table[Binomial[2 n, n] - (2^n + Binomial[2 n, n])/2, {n, 30}] (* Vincenzo Librandi, Feb 01 2016 *)
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PROG
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(Magma) [Binomial(2*n, n)-(2^n+Binomial(2*n, n))/2: n in [1..30]]; // Vincenzo Librandi, Feb 01 2016
(PARI) a(n)=binomial(2*n-1, n-1)-2^(n-1) \\ Charles R Greathouse IV, Feb 01 2016
(Python)
from math import comb
def A108958(n): return comb((n<<1)-1, n-1)-(1<<n-1) # Chai Wah Wu, Sep 23 2022
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CROSSREFS
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Cf. A143418, A005317.
Sequence in context: A221863 A216263 A003517 * A005284 A198694 A220101
Adjacent sequences: A108955 A108956 A108957 * A108959 A108960 A108961
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KEYWORD
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easy,nonn
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AUTHOR
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Jeffrey Shallit, Jul 22 2005
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STATUS
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approved
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