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A101785 G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2).

%I A101785 #58 Aug 05 2024 05:18:55
%S A101785 1,1,1,2,5,12,30,79,213,584,1628,4600,13138,37871,110043,321978,
%T A101785 947813,2805104,8341608,24912004,74686460,224694128,678143656,
%U A101785 2052640752,6229616730,18952875247,57792705415,176596786934,540679385663
%N A101785 G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2).
%C A101785 Formula may be derived using the Lagrange Inversion theorem (cf. A049124).
%C A101785 a(n) = number of Dyck n-paths (A000108) all of whose descents have odd length. For example, a(3) counts UUUDDD, UDUDUD. - _David Callan_, Jul 25 2005
%C A101785 The number of noncrossing partitions of [n] with all blocks of odd size. E.g.: a(4)=5 with the five partitions being 123/4, 124/3, 134/2,1/234 and 1/2/3/4. - _Louis_ Shapiro, Jan 07 2006
%C A101785 Number of ordered trees with n edges in which every non-leaf vertex has an odd number of children. - _David Callan_, Mar 30 2007
%C A101785 Number of valid hook configurations of permutations of [n] that avoid the patterns 312 and 321. - _Colin Defant_, Apr 28 2019
%H A101785 Alois P. Heinz, <a href="/A101785/b101785.txt">Table of n, a(n) for n = 0..1000</a>
%H A101785 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H A101785 Colin Defant, <a href="http://arxiv.org/abs/1904.10451">Motzkin intervals and valid hook configurations</a>, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
%H A101785 Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H A101785 Vaclav Kotesovec, <a href="http://oeis.org/A212382/a212382.pdf">Asymptotic of subsequences of A212382</a>
%H A101785 Helmut Prodinger, <a href="https://arxiv.org/abs/2408.01290">Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences</a>, arXiv:2408.01290 [math.CO], 2024.
%F A101785 a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(n, 2*k)/(2*k+1) for n>0, with a(0)=1.
%F A101785 G.f.: (1/x) * Series_Reversion( x*(1-x^2)/(1+x-x^2) ).
%F A101785 Recurrence: 4*n*(n+1)*(91*n^2 - 379*n + 360)*a(n) = 6*n*(182*n^3 - 849*n^2 + 1075*n - 264)*a(n-1) - 2*(182*n^4 - 1122*n^3 + 2011*n^2 - 603*n - 648)*a(n-2) + 6*(n-3)*(364*n^3 - 1698*n^2 + 2267*n - 696)*a(n-3) - 5*(n-4)*(n-3)*(91*n^2 - 197*n + 72)*a(n-4). - _Vaclav Kotesovec_, Sep 17 2013
%F A101785 a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3/4 + 1/(4*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) + 1/2*sqrt(19/6 + 76/(3*(4103 + 273*sqrt(273))^(1/3)) - 1/6*(4103 + 273*sqrt(273))^(1/3) + 63/2*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) = 3.228704951094501729... is the root of the equation 5 - 24*d + 4*d^2 - 12*d^3 + 4*d^4 = 0 and c = 0.82499074317860885542266460957609663272... is the root of the equation -125 - 3376*c^2 - 22080*c^4 - 23296*c^6 + 93184*c^8 = 0. - _Vaclav Kotesovec_, added Sep 17 2013, updated Jan 04 2014
%F A101785 G.f.: 1/(9*(3-3*x+x^2))*(x^2+27- x^2*(2*x+3)^3*(x-6)^3/(9*(3-3*x+x^2)^3*S(0) - x^2*(2*x+3)^2*(x-6)^2 )), where S(k) = 4*k+3 - x^2*(2*x^2-9*x-18)^2*(3*k+4)*(6*k+5)/( 18*(4*k+5)*(3-3*x+x^2)^3 - x^2*(2*x^2-9*x-18)^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 26 2013
%e A101785 Generated from Fibonacci polynomials (A011973) and
%e A101785 coefficients of odd powers of 1/(1-x):
%e A101785 a(1) = 1*1/1
%e A101785 a(2) = 1*1/1 + 0*1/3
%e A101785 a(3) = 1*1/1 + 1*3/3
%e A101785 a(4) = 1*1/1 + 2*6/3 + 0*1/5
%e A101785 a(5) = 1*1/1 + 3*10/3 + 1*5/5
%e A101785 a(6) = 1*1/1 + 4*15/3 + 3*15/5 + 0*1/7
%e A101785 a(7) = 1*1/1 + 5*21/3 + 6*35/5 + 1*7/7
%e A101785 a(8) = 1*1/1 + 6*28/3 + 10*70/5 + 4*28/7 + 0*1/9
%e A101785 This process is equivalent to the formula:
%e A101785 a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1,k)*C(n,2*k)/(2*k+1).
%t A101785 Flatten[{1,Table[Sum[Binomial[n-k-1,k]*Binomial[n,2*k]/(2*k+1),{k,0,Floor[(n-1)/2]}],{n,1,20}]}] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%t A101785 CoefficientList[InverseSeries[Series[x*(1-x^2)/(1+x-x^2), {x, 0, 30}], x]/x, x] (* _G. C. Greubel_, May 03 2019 *)
%o A101785 (PARI) {a(n)=if(n==0,1,sum(k=0,(n-1)\2,binomial(n-k-1,k)*binomial(n,2*k)/(2*k+1)))}
%o A101785 for(n=1, 40, print1(a(n), ", "))
%o A101785 (PARI) N=66; Vec(serreverse(x/(1+sum(k=1,N,x^(2*k-1)))+O(x^N))/x) /* _Joerg Arndt_, Aug 19 2012 */
%o A101785 (Magma) [n eq 0 select 1 else (&+[Binomial(n-k-1,k)*Binomial(n, 2*k)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [0..30]]; // _G. C. Greubel_, May 03 2019
%o A101785 (Sage) [1]+[sum(binomial(n-k-1, k)*binomial(n, 2*k)/(2*k+1) for k in (0..floor((n-1)/2))) for n in (1..30)] # _G. C. Greubel_, May 03 2019
%Y A101785 Cf. A011973, A049124.
%Y A101785 Column k=2 of A212382.
%K A101785 nonn
%O A101785 0,4
%A A101785 _Paul D. Hanna_, Dec 16 2004