OFFSET
0,2
COMMENTS
If Y is a fixed 2-subset of a (2n+1)-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
REFERENCES
V. N. Smith and L. Shapiro, Catalan numbers, Pascal's triangle and mutators, Congressus Numerant., 205 (2010), 187-197.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
C. Bean, M. Tannock and H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015.
Milan Janjic, Two Enumerative Functions
Mark C. Wilson, Asymptotics for generalized Riordan arrays, International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005. (However, the asymptotics given there on p. 328 for a(n) give different results for me. - Ralf Stephan, Dec 28 2013)
FORMULA
a(n) = 3 * binomial(2n-1, n) (n>0). - Len Smiley, Nov 03 2001
a(n) = 3*A001700(n-1), (n>=1).
G.f.: (1+xC(x))/(1-2xC(x)), C(x) the g.f. of A000108. - Paul Barry, Dec 17 2004
a(n) = A003409(n), n>0. - R. J. Mathar, Oct 23 2008
a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n))-0^n/2. - Peter Luschny, Dec 16 2015
a(n) ~ (3/2)*4^n*(1-(1/8)/n+(1/128)/n^2+(5/1024)/n^3-(21/32768)/n^4)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
a(n) = 2^(1-n)*Sum_{k=0..n}(binomial(k+n,k)*binomial(2*n-1,n-k))), n>0, a(0)=1. - Vladimir Kruchinin, Nov 23 2016
E.g.f.: (3*exp(2*x)*BesselI(0,2*x) - 1)/2. - Ilya Gutkovskiy, Nov 23 2016
a(n) = [x^n] C(-x)^(-3*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Oct 16 2024
MAPLE
a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n))-0^n/2;
seq(simplify(a(n)), n=0..24); # Peter Luschny, Dec 16 2015
MATHEMATICA
Join[{1}, Table[3*Binomial[2n-1, n], {n, 30}]] (* Harvey P. Dale, Aug 11 2015 *)
PROG
(PARI) concat([1], for(n=1, 50, print1(3*binomial(2*n-1, n), ", "))) \\ G. C. Greubel, Jan 23 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved