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A052510 Number of labeled planar binary trees with 2n-1 elements (external nodes or internal nodes).

%I A052510 #69 Dec 04 2022 08:32:21
%S A052510 1,6,240,25200,5080320,1676505600,821966745600,560992303872000,
%T A052510 508633022177280000,591438478187741184000,858123464716031754240000,
%U A052510 1519736656012092236759040000,3226517823533365056503808000000,8089341114715793820234547200000000
%N A052510 Number of labeled planar binary trees with 2n-1 elements (external nodes or internal nodes).
%C A052510 Equals central terms of A174449. - _G. C. Greubel_, Nov 29 2021
%H A052510 G. C. Greubel, <a href="/A052510/b052510.txt">Table of n, a(n) for n = 1..200</a>
%H A052510 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=54">Encyclopedia of Combinatorial Structures 54</a> [Link is broken.]
%F A052510 E.g.f.: ((1/2)/x)*(1-sqrt(1-4*x^2)). [With interspersed zeros.]
%F A052510 Recurrence: b(1)=1, b(2)=0, b(n)=(4*n^3-12*n^2+8*n)*b(n-2)/(n+1) and a(n) = b(2*n-1).
%F A052510 a(n) = (2n-1)/n * ( (2(n-1))! / (n-1)! )^2. - Travis Kowalski (kowalski(AT)euclid.UCSD.Edu), Dec 15 2000
%F A052510 i*sin(arcsec(2*x)) = -1/(2*x) + x + 6*x^3/3! + 240*x^5/5! + 25200*x^7/7! + ...
%F A052510 a(n) = 2^(n-1) * A036770(n).
%F A052510 a(n) = (2*n-1)! * A000108(n-1). - _Michail Stamatakis_, Jan 24 2019
%F A052510 Sum_{n>=1} 1/a(n) = 1 + StruveL(0, 1/2)*Pi/8 + StruveL(1, 1/2)*Pi/4, where StruveL is the modified Struve function. - _Amiram Eldar_, Dec 04 2022
%p A052510 spec := [S, {S=Union(Z, Prod(Z, S, S))}, labeled]:
%p A052510 seq(combstruct[count](spec, size=2*n-1), n=1..14);
%p A052510 # second Maple program:
%p A052510 a:= proc(n) option remember; `if`(n<2, n,
%p A052510       4*(n-1)*(2*n-3)*(2*n-1)*a(n-1)/n)
%p A052510     end:
%p A052510 seq(a(n), n=1..22);  # _Alois P. Heinz_, Dec 03 2019
%t A052510 nn=20;f[x_]:=Sum[a[n]x^n/n!,{n,0,nn}];s=SolveAlways[0==Series[f[x]-x(1+f[x]^2),{x,0,nn}],x];Select[Flatten[Table[a[n],{n,0,nn}]/.s],#>0&]  (* _Geoffrey Critzer_, Mar 23 2013 *)
%t A052510 RecurrenceTable[{a[1]==1, n*a[n]==(4*(n-1)*(2*n-3)*(2*n-1))*a[n-1]}, a[n], {n,1,22}] (* _Georg Fischer_, Dec 03 2019 following _Alois P. Heinz_ *)
%t A052510 a[n_]:= CatalanNumber[n-1] Gamma[2n]; Array[a,14] (* _Peter Luschny_, Dec 03 2019 *)
%o A052510 (PARI) a=vector(28);print1(a[1]=1,", ");forstep(k=1,#a-2,2,print1(a[k+2]=4*a[k]*(k^3+3*k^2+2*k)/(k+3),", ")) \\ _Hugo Pfoertner_, Dec 04 2019
%o A052510 (Sage) [factorial(2*n-1)*catalan_number(n-1) for n in (1..15)] # _G. C. Greubel_, Nov 29 2021
%o A052510 (Magma) [Factorial(2*n-1)*Catalan(n-1): n in [1..15]]; // _G. C. Greubel_, Nov 29 2021
%Y A052510 Cf. A000108, A036770, A101921, A174449.
%K A052510 easy,nonn
%O A052510 1,2
%A A052510 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052510 Edited by _Georg Fischer_, Dec 03 2019