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A079489 a(n) = (2*4^n*binomial(2*n, n) - binomial(4*n + 1, 2*n)) / (n + 1).

%I A079489 #97 Dec 21 2024 07:49:00
%S A079489 1,3,22,211,2306,27230,338444,4362627,57788170,781825066,10757497972,
%T A079489 150073096238,2117778107732,30176799215196,433586825237912,
%U A079489 6274885068167651,91383942213277530,1338275570267001458,19695358741104824036,291137841642777382330,4320734864185863437820
%N A079489 a(n) = (2*4^n*binomial(2*n, n) - binomial(4*n + 1, 2*n)) / (n + 1).
%C A079489 a(n) is the number of ordered trees on 2n-1 edges in which every subtree of the root (including its rooting edge) has an even number of edges, except for the leftmost subtree which has an odd number of edges (including its rooting edge). - _David Callan_, Apr 10 2012
%C A079489 a(n) is the number of 2 X 2n Young tableaux with a wall between the first and second row in each even column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021].
%C A079489   Example for a(1)=3:
%C A079489    3 4  2 4  2 3
%C A079489      -    -    -
%C A079489    1 2, 1 3, 1 4. - _Michael Wallner_, Mar 09 2022
%H A079489 G. C. Greubel, <a href="/A079489/b079489.txt">Table of n, a(n) for n = 0..830</a>
%H A079489 Cyril Banderier and Michael Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47.html">Young Tableaux with Periodic Walls: Counting with the Density Method</a>, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
%H A079489 Feihu Liu and Guoce Xin, <a href="https://arxiv.org/abs/2401.14627">Simple Generating Functions for Certain Young Tableaux with Periodic Walls</a>, arXiv:2401.14627 [math.CO], 2024.
%H A079489 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344 (Table A.1).
%H A079489 Alon Regev, <a href="http://arxiv.org/abs/1208.3915">Enumerating triangulations by parallel diagonals</a>, arXiv:1208.3915 [math.CO], 2012. - _N. J. A. Sloane_, Dec 25 2012
%H A079489 Alon Regev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Regev/alon3.html">Enumerating Triangulations by Parallel Diagonals</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.5.
%F A079489 Series reversion of x(1-x^2)/(1+x^2)^2 expanded in odd powers of x. [Previous name.]
%F A079489 If x = y*(1-y^2)/(1+y^2)^2 then y = x + 3*x^3 + 22*x^5 + 211*x^7 + 2306*x^9 + ...
%F A079489 G.f. A(x) satisfies x*A(x^2) = (C(x) - C(-x))/(C(x) + C(-x)) where C(x) is g.f. of the Catalan numbers A000108.
%F A079489 a(n) = Sum_{k=0..2n} (-1)^k * A000108(2*n-k) * A000108(k). - _David Callan_, Aug 16 2006
%F A079489 a(n) = ((2^(4n+2))/Gamma(1/2)) * ((Gamma(n+1/2)/(2*Gamma(n+2))) - Gamma(2n+3/2)/Gamma(2n+3)). [David Dickson (dcmd(AT)unimelb.edu.au), Nov 10 2009]
%F A079489 G.f.: exp( Sum_{n>=1} C(4n-1,2n)*x^n/n ). - _Paul D. Hanna_, Dec 30 2010
%F A079489 G.f.: C(sqrt(x))*C(-sqrt(x)) where C(x) is the g.f. for the Catalan numbers A000108. - _David Callan_, Apr 10 2012
%F A079489 D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -2*n*(32*n^2-32*n+11)*a(n-1) +16*(4*n-5)*(4*n-3)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Nov 29 2012
%F A079489 a(n) ~ (2-sqrt(2))*16^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 20 2013
%F A079489 a(n) = 2^(2*n+1)*Catalan(n) - Catalan(2*n+1) (see Regev). It follows that the 2-adic valuations of a(n) and Catalan(n) are equal. In particular, a(n) is odd iff n is of the form 2^m - 1. - _Peter Bala_, Aug 02 2016
%F A079489 G.f.: (sqrt(2) * sqrt(1 + sqrt(1-16*x)) - sqrt(1-16*x) - 1)/(4*x). - _Vladimir Reshetnikov_, Sep 25 2016
%F A079489 G.f. A(x) satisfies A(x^2) = C(x)^2*r(-x*C(x)^2), where C(x) is g.f. of the Catalan numbers A000108, and r(x) is g.f. of the large Schröder numbers A006318. - _Alexander Burstein_, Nov 21 2019
%F A079489 From _Peter Bala_, Sep 14 2021: (Start)
%F A079489 A(x) = exp( Sum_{n >= 1} (1/2)*binomial(4*n,2*n)*x^n/n ).
%F A079489 1 + x*A(x) is the o.g.f. of A066357.
%F A079489 The sequence defined by b(n) := [x^n] A(x)^n begins [1, 3, 53, 1056, 22181, 480003, 10588508, 236720424, ...]  and satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 3. See A333563. Cf. A060941. (End)
%F A079489 From _Peter Bala_, Oct 23 2024: (Start)
%F A079489 For integer r and positive integer s, define a sequence {u(n) : n >= 0} by setting u(n) = [x^(s*n)] A(x)^(r*n). We conjecture that the supercongruence u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and for all positive integers n and k.
%F A079489 Let B(x) = 1/x * series_reversion(x*A(x)). Define a sequence {v(n) : n >= 0} by setting v(n) = [x^(s*n)] B(x)^(r*n). We conjecture that the supercongruence v(n*p^k) == v(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and for all positive integers n and k. (End)
%p A079489 a := n -> (2*4^n*binomial(2*n, n) - binomial(4*n + 1, 2*n)) / (n + 1):
%p A079489 seq(a(n), n = 0..20);  # _Peter Luschny_, Aug 26 2024
%t A079489 ((Sqrt[2] Sqrt[1 + Sqrt[1 - 16 x]] - Sqrt[1 - 16 x] - 1)/(4 x) + O[x]^20)[[3]] (* _Vladimir Reshetnikov_, Sep 25 2016 *)
%t A079489 CoefficientList[Series[-(1 - Sqrt[1 - 4*Sqrt[x]])*(1 - Sqrt[1 + 4*Sqrt[x]])/(4*x), {x,0,50}], x] (* _G. C. Greubel_, Apr 13 2017 *)
%o A079489 (PARI) a(n)=if(n<0,0,polcoeff(serreverse(x*(1-x^2)/(1+x^2)^2+O(x^(2*n+3))),2*n+1))
%o A079489 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,binomial(4*m-1,2*m)*x^m/m)+x*O(x^n)),n)} \\ _Paul D. Hanna_, Dec 30 2010
%Y A079489 Final diagonal of triangle in A078990.
%Y A079489 Cf. A000108, A006318, A066357, A060941, A066357, A300386 - A300389, A333563.
%K A079489 nonn,easy
%O A079489 0,2
%A A079489 _N. J. A. Sloane_, Jan 20 2003
%E A079489 New name by _Peter Luschny_, Aug 26 2024