cp's OEIS Frontend

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

%I A029651 #75 Jul 22 2025 13:58:53
%S A029651 1,3,9,30,105,378,1386,5148,19305,72930,277134,1058148,4056234,
%T A029651 15600900,60174900,232676280,901620585,3500409330,13612702950,
%U A029651 53017895700,206769793230,807386811660,3156148445580,12350146091400,48371405524650
%N A029651 Central elements of the (1,2)-Pascal triangle A029635.
%C A029651 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
%D A029651 V. N. Smith and L. Shapiro, Catalan numbers, Pascal's triangle and mutators, Congressus Numerant., 205 (2010), 187-197.
%H A029651 G. C. Greubel, <a href="/A029651/b029651.txt">Table of n, a(n) for n = 0..1000</a>
%H A029651 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H A029651 C. Bean, M. Tannock and H. Ulfarsson, <a href="http://arxiv.org/abs/1512.08155">Pattern avoiding permutations and independent sets in graphs</a>, arXiv:1512.08155 [math.CO], 2015.
%H A029651 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A029651 Mark C. Wilson, <a href="https://doi.org/10.46298/dmtcs.3389">Asymptotics for generalized Riordan arrays</a>, 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)
%F A029651 a(n) = 3 * binomial(2n-1, n) (n>0). - _Len Smiley_, Nov 03 2001
%F A029651 a(n) = 3*A001700(n-1), (n>=1).
%F A029651 G.f.: (1+xC(x))/(1-2xC(x)), C(x) the g.f. of A000108. - _Paul Barry_, Dec 17 2004
%F A029651 a(n) = A003409(n), n>0. - _R. J. Mathar_, Oct 23 2008
%F A029651 a(n) = Sum_{k=0..n} A039599(n,k)*A000034(k). - _Philippe Deléham_, Oct 29 2008
%F A029651 a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n))-0^n/2. - _Peter Luschny_, Dec 16 2015
%F A029651 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
%F A029651 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
%F A029651 E.g.f.: (3*exp(2*x)*BesselI(0,2*x) - 1)/2. - _Ilya Gutkovskiy_, Nov 23 2016
%F A029651 a(n) = A143398(2n,n) = A145460(2n,n). - _Alois P. Heinz_, Sep 09 2018
%F A029651 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
%p A029651 a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n))-0^n/2;
%p A029651 seq(simplify(a(n)), n=0..24); # _Peter Luschny_, Dec 16 2015
%t A029651 Join[{1},Table[3*Binomial[2n-1,n],{n,30}]] (* _Harvey P. Dale_, Aug 11 2015 *)
%o A029651 (PARI) concat([1], for(n=1, 50, print1(3*binomial(2*n-1,n), ", "))) \\ _G. C. Greubel_, Jan 23 2017
%Y A029651 Cf. A001700, A143398, A145460.
%Y A029651 Essentially a duplicate of A003409.
%K A029651 nonn,easy
%O A029651 0,2
%A A029651 _Mohammad K. Azarian_
%E A029651 More terms from _David W. Wilson_