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
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.
FORMULA
a(n) = Sum_{k=0..n} binomial(binomial(n, k), 2).
From Vladeta Jovovic, Jul 24 2005: (Start)
E.g.f.: exp(2*x)*(BesselI(0, 2*x)-1)/2. (End)
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) = 1/2*Sum_{k=1..n} binomial(n,k)^2 - binomial(n,k). - Gerry Martens, Oct 09 2022
a(n) ~ 2^(2*n-1)/sqrt(n*Pi). - Stefano Spezia, Apr 17 2024
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
(Python)
from math import comb
def A108958(n): return comb((n<<1)-1, n-1)-(1<<n-1) # Chai Wah Wu, Sep 23 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jeffrey Shallit, Jul 22 2005
STATUS
approved