A029651 Central elements of the (1,2)-Pascal triangle A029635.

1, 3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

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)

Crossrefs

Cf. A001700, A143398, A145460.

Essentially a duplicate of A003409.

Programs

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 *)

PARI

concat([1], for(n=1, 50, print1(3*binomial(2*n-1,n), ", "))) \\ G. C. Greubel, Jan 23 2017

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) = Sum_{k=0..n} A039599(n,k)*A000034(k). - Philippe Deléham, Oct 29 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) = A143398(2n,n) = A145460(2n,n). - Alois P. Heinz, Sep 09 2018

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

Extensions

More terms from David W. Wilson