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

0,2

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) = 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

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

Cf. A001700, A143398, A145460.

Sequence in context: A148956 A339036 A360715 * A003409 A316371 A181933

Adjacent sequences: A029648 A029649 A029650 * A029652 A029653 A029654

Keyword

nonn

Author

Mohammad K. Azarian

Extensions

More terms from David W. Wilson

Status

approved