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

1, 6, 240, 25200, 5080320, 1676505600, 821966745600, 560992303872000, 508633022177280000, 591438478187741184000, 858123464716031754240000, 1519736656012092236759040000, 3226517823533365056503808000000, 8089341114715793820234547200000000

Offset: 1

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Equals central terms of A174449. - G. C. Greubel, Nov 29 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 1..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 54 [Link is broken.]

Crossrefs

Cf. A000108, A036770, A101921, A174449.

Programs

Magma

[Factorial(2*n-1)*Catalan(n-1): n in [1..15]]; // G. C. Greubel, Nov 29 2021

Maple

spec := [S, {S=Union(Z, Prod(Z, S, S))}, labeled]:
seq(combstruct[count](spec, size=2*n-1), n=1..14);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
      4*(n-1)*(2*n-3)*(2*n-1)*a(n-1)/n)
    end:
seq(a(n), n=1..22);  # Alois P. Heinz, Dec 03 2019

Mathematica

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 *)
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 *)
a[n_]:= CatalanNumber[n-1] Gamma[2n]; Array[a,14] (* Peter Luschny, Dec 03 2019 *)

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

Sage

[factorial(2*n-1)*catalan_number(n-1) for n in (1..15)] # G. C. Greubel, Nov 29 2021

Formula

E.g.f.: ((1/2)/x)*(1-sqrt(1-4*x^2)). [With interspersed zeros.]

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).

a(n) = (2n-1)/n * ( (2(n-1))! / (n-1)! )^2. - Travis Kowalski (kowalski(AT)euclid.UCSD.Edu), Dec 15 2000

i*sin(arcsec(2*x)) = -1/(2*x) + x + 6*x^3/3! + 240*x^5/5! + 25200*x^7/7! + ...

a(n) = 2^(n-1) * A036770(n).

a(n) = (2*n-1)! * A000108(n-1). - Michail Stamatakis, Jan 24 2019

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

Extensions

Edited by Georg Fischer, Dec 03 2019