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A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.

%I A014300 #122 Nov 25 2023 07:35:07
%S A014300 1,2,7,24,86,314,1163,4352,16414,62292,237590,909960,3497248,13480826,
%T A014300 52097267,201780224,783051638,3044061116,11851853042,46208337584,
%U A014300 180383564228,704961896036,2757926215742,10799653176704,42326626862636,166021623024584,651683311373788
%N A014300 Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.
%C A014300 Also total number of blocks of odd size in all Catalan(n) possible noncrossing partitions of [n].
%C A014300 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).
%C A014300 Row sums of A119307. - _Paul Barry_, May 13 2006
%C A014300 Hankel transform is A079935. - _Paul Barry_, Jul 17 2009
%C A014300 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
%C A014300 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
%C A014300 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
%C A014300 The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p >= 3 and positive integers n and k. - _Peter Bala_, Jan 07 2022
%H A014300 Alois P. Heinz, <a href="/A014300/b014300.txt">Table of n, a(n) for n = 1..500</a>
%H A014300 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A014300 Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://lrecits.usthb.dz/6.5.pdf">Diagonal sums in Pascal pyramid (1, 2, r)</a>, Les Annales RECITS (2019) Vol. 6, 45-52.
%H A014300 N. Dershowitz and S. Zaks, <a href="http://dx.doi.org/10.1016/0012-365X(86)90120-2">Ordered trees and non-crossing partitions</a>, Discrete Math., 62 (1986), 215-218.
%H A014300 Emeric Deutsch and L. Shapiro, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%H A014300 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A014300 a(n) = (binomial(2*n, n) + A000957(n))/3; [simplified by _Alexander Burstein_, Nov 24 2023]
%F A014300 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k-1). - _Vladeta Jovovic_, Aug 28 2002
%F A014300 G.f.: 2*z/(1-4*z+(1+2*z)*sqrt(1-4*z)).
%F A014300 a(n) = Sum_{j=0..floor((n-1)/2)} binomial(2*n-2*j-2, n-1).
%F A014300 2*a(n) + a(n-1) = (3*n-1)*Catalan(n-1). - _Vladeta Jovovic_, Dec 03 2004
%F A014300 a(n) = (-1)^n*Sum_{i=0..n} Sum_{j=n..2*n} (-1)^(i+j)*binomial(j, i). - _Benoit Cloitre_, Jun 18 2005
%F A014300 a(n) = Sum_{k=0..n} C(2*k,n) [offset 0]. - _Paul Barry_, May 13 2006
%F A014300 a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n+k-1,k-1). - _Paul Barry_, Jul 18 2006
%F A014300 From _Paul Barry_, Jul 17 2009: (Start)
%F A014300 a(n) = Sum_{k=0..n} C(2*n-k,n-k)*(1+(-1)^k)/2.
%F A014300 a(n) = Sum_{k=0..n} C(n+k,k)*(1+(-1)^(n-k))/2. (End)
%F A014300 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
%F A014300 a(n) = -binomial(2*n,n-1)*hyper2F1([1,2*n+1],[n+2], 2). - _Peter Luschny_, Jul 25 2014
%F A014300 a(n) = [x^n] x/((1 - x^2)*(1 - x)^n). - _Ilya Gutkovskiy_, Oct 25 2017
%F A014300 a(n) ~ 4^n / (3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 25 2017
%F A014300 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
%F A014300 a(n) = A333564(n)/2^n. - _Peter Bala_, Apr 09 2020
%F A014300 a(n) = (1/2)*(binomial(2*n,n) - A072547(n)). - _Peter Bala_, Mar 28 2023
%p A014300 a:= proc(n) a(n):= `if`(n<3, n, ((12-40*n+21*n^2) *a(n-1)+
%p A014300        2*(3*n-1)*(2*n-3) *a(n-2))/ (2*(3*n-4)*n))
%p A014300     end:
%p A014300 seq(a(n), n=1..30);  # _Alois P. Heinz_, Oct 30 2012
%t A014300 Rest[CoefficientList[Series[2x/(1-4x+(1+2x)Sqrt[1-4x]),{x,0,40}],x]]  (* _Harvey P. Dale_, Apr 25 2011 *)
%t A014300 a[n_] := Sum[Binomial[2k, n-1], {k, 0, n-1}]; Array[a, 30] (* _Jean-François Alcover_, Dec 25 2015, after _Paul Barry_ *)
%o A014300 (PARI) a(n) = n--; sum(k=0, n, binomial(2*k,n)); \\ _Michel Marcus_, May 18 2018
%o A014300 (Magma) [(&+[(-1)^(n-k)*Binomial(n+k-1, k-1): k in [0..n]]): n in [1..30]]; // _G. C. Greubel_, Feb 19 2019
%o A014300 (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
%o A014300 (Python)
%o A014300 from itertools import count, islice
%o A014300 def A014300_gen(): # generator of terms
%o A014300     yield from (1,2)
%o A014300     a, c = 1, 1
%o A014300     for n in count(1):
%o A014300         yield (a:=(3*n+5)*(c:=c*((n<<2)+2)//(n+2))-a>>1)
%o A014300 A014300_list = list(islice(A014300_gen(),20)) # _Chai Wah Wu_, Apr 26 2023
%Y A014300 Cf. A059481, A000957, A000984, A072547, A119259, A119307, A079935, A333564.
%K A014300 nonn,nice,easy
%O A014300 1,2
%A A014300 _Emeric Deutsch_