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A108958 Number of unordered pairs of distinct length-n binary words having the same number of 1's. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..26.

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.

FORMULA

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

EXAMPLE

a(3) = 6 because the pairs are {001,010}, {001,100}, {010,100}, {011,101}, {011,110}, {101,110}.

MAPLE

with(combinat) a:= proc(n) add(binomial(binomial(n, k), 2), k=0..n) end;

MATHEMATICA

Table[Binomial[2 n, n] - (2^n + Binomial[2 n, n])/2, {n, 30}] (* Vincenzo Librandi, Feb 01 2016 *)

PROG

(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

CROSSREFS

Cf. A143418, A005317.

Sequence in context: A221863 A216263 A003517 * A005284 A198694 A220101

Adjacent sequences:  A108955 A108956 A108957 * A108959 A108960 A108961

KEYWORD

easy,nonn

AUTHOR

Jeffrey Shallit, Jul 22 2005

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.