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 A029651 Central elements of the (1,2)-Pascal triangle A029635. 12
 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 (list; graph; refs; listen; history; text; internal format)
 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: A145268 A148956 A339036 * A003409 A316371 A181933 Adjacent sequences:  A029648 A029649 A029650 * A029652 A029653 A029654 KEYWORD nonn AUTHOR EXTENSIONS More terms from David W. Wilson STATUS approved

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Last modified December 4 19:01 EST 2020. Contains 338936 sequences. (Running on oeis4.)