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 A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges. 25
 1, 2, 7, 24, 86, 314, 1163, 4352, 16414, 62292, 237590, 909960, 3497248, 13480826, 52097267, 201780224, 783051638, 3044061116, 11851853042, 46208337584, 180383564228, 704961896036, 2757926215742, 10799653176704, 42326626862636, 166021623024584, 651683311373788 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also total number of blocks of odd size in all Catalan(n) possible noncrossing partitions of [n]. Convolution of the sequence of central binomial coefficients 1,2,6,20,70,... (A000984) and of the sequence of Fine numbers 1,0,1,2,6,18,... (A000957). Row sums of A119307. - Paul Barry, May 13 2006 Hankel transform is A079935. - Paul Barry, Jul 17 2009 Also for n>=1 the number of unimodal functions f:[n]->[n] with f(i)<>f(i+1). a(3) = 7: [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,3,1], [2,3,2], [3,2,1]. - Alois P. Heinz, May 23 2013 Also, number of sets of n rational numbers on [0,1) such that if x belongs to the set, the fractional part of 2x also belongs to it. - Jianing Song and Andrew Howroyd, May 18 2018 Let A(i, j) denote the infinite array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i times to the function ((-1)^(n + 1) + 1)/2 for n > 0. Then A(n, n) equals a(n) for all n > 0. - John M. Campbell, Jan 20 2019 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..500 Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5. Hacène Belbachir, Abdelghani Mehdaoui, Diagonal sums in Pascal pyramid (1, 2, r), Les Annales RECITS (2019) Vol. 6, 45-52. N. Dershowitz and S. Zaks, Ordered trees and non-crossing partitions, Discrete Math., 62 (1986), 215-218. E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265. FORMULA a(n) = (2*binomial(2*n-1, n) + A000957(n))/3; a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k-1). - Vladeta Jovovic, Aug 28 2002 G.f.: 2*z/(1-4*z+(1+2*z)*sqrt(1-4*z)). a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-2, n-1). 2*a(n) + a(n-1) = (3*n-1)*Catalan(n-1). - Vladeta Jovovic, Dec 03 2004 a(n) = (-1)^n*Sum_{i=0..n} Sum_{j=n..2*n} (-1)^(i+j)*binomial(j, i). - Benoit Cloitre, Jun 18 2005 a(n) = Sum_{k=0..n} C(2*k,n) [offset 0]. - Paul Barry, May 13 2006 a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n+k-1,k-1). - Paul Barry, Jul 18 2006 From Paul Barry, Jul 17 2009: (Start) a(n) = Sum_{k=0..n} C(2*n-k,n-k)*(1+(-1)^k)/2. a(n) = Sum_{k=0..n} C(n+k,k)*(1+(-1)^(n-k))/2. (End) a(n) is the coefficient of x^(n+1)*y^(n+1) in 1/(1- x^2*y/((1-2*x)*(1-y))). - Ira M. Gessel, Oct 30 2012 a(n) = -binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2], 2). - Peter Luschny, Jul 25 2014 a(n) = [x^n] x/((1 - x^2)*(1 - x)^n). - Ilya Gutkovskiy, Oct 25 2017 a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 25 2017 D-finite with recurrence: 2*n*a(n) +(-3*n-4)*a(n-1) +2*(-9*n+19)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 20 2020 a(n) = A333564(n)/2^n. - Peter Bala, Apr 09 2020 MAPLE a:= proc(n) a(n):= `if`(n<3, n, ((12-40*n+21*n^2) *a(n-1)+        2*(3*n-1)*(2*n-3) *a(n-2))/ (2*(3*n-4)*n))     end: seq(a(n), n=1..30);  # Alois P. Heinz, Oct 30 2012 MATHEMATICA Rest[CoefficientList[Series[2x/(1-4x+(1+2x)Sqrt[1-4x]), {x, 0, 40}], x]]  (* Harvey P. Dale, Apr 25 2011 *) a[n_] := Sum[Binomial[2k, n-1], {k, 0, n-1}]; Array[a, 30] (* Jean-François Alcover, Dec 25 2015, after Paul Barry *) PROG (PARI) a(n) = n--; sum(k=0, n, binomial(2*k, n)); \\ Michel Marcus, May 18 2018 (MAGMA) [(&+[(-1)^(n-k)*Binomial(n+k-1, k-1): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Feb 19 2019 (Sage) [sum((-1)^(n-k)*binomial(n+k-1, k-1) for k in (0..n)) for n in (1..30)] # G. C. Greubel, Feb 19 2019 CROSSREFS Cf. A059481, A000957, A000984, A119307, A079935, A333564. Sequence in context: A052986 A053368 A141753 * A128086 A131824 A256938 Adjacent sequences:  A014297 A014298 A014299 * A014301 A014302 A014303 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified October 23 20:03 EDT 2020. Contains 337975 sequences. (Running on oeis4.)