A036771 -id:A036771 - cp's OEIS frontend

A036770 Number of labeled rooted trees with a degree constraint: (2n)!/(2^n) C(2*n+1, n).

1, 3, 60, 3150, 317520, 52390800, 12843230400, 4382752374000, 1986847742880000, 1155153277710432000, 838011196011749760000, 742058914068404412480000, 787724078011075453248000000, 987468397792455300321600000000, 1443283810213452666950050560000000

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of rooted labeled strictly binary trees (each vertex has exactly two children or none) on 2*n + 1 vertices. - Geoffrey Critzer, Nov 13 2011

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 46.
Lajos Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10; see Eqs. (12) and (13) on p. 4.
Eric Weisstein's World of Mathematics, Strongly Binary Tree.
Index entries for sequences related to rooted trees

Cf. A001190, A036771, A036772, A036773, A052510.

[Factorial(2*n)/(2^n) * Binomial(2*n+1, n): n in [0..15]]; // Vincenzo Librandi, Jan 29 2020

spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},labeled]: seq(combstruct[count](spec,size=n)

Range[0, 19]! CoefficientList[Series[(1 - (1 - 2 x^2)^(1/2))/x, {x, 0, 20}], x] (* Geoffrey Critzer, Nov 13 2011 *)

Recurrence (with interpolated zeros): Define (b(n): n >= 0) by b(2*m+1) = a(m) and b(2*m) = 0 for m >= 0. Then the sequence (b(n): n >= 0) satisfies the recurrence (-2*n^3 - 6*n^2 - 4*n)*b(n) + (n + 3)*b(n+2) = 0 for n >= 0 with b(0) = 0 and b(1) = 1. [Corrected by Petros Hadjicostas, Jun 07 2019]
E.g.f. with interpolated zeros: G(x) = Sum_{n >= 0} b(n)*x^n/n! = Sum_{m >= 0} a(m)*x^(2*m + 1)/(2*m + 1)! = 1/x * (1 - (1 - 2*x^2)^(1/2)) for 0 < |x| < 1/sqrt(2). [Edited by Petros Hadjicostas, Jun 07 2019]
E.g.f. with interpolated zeros satisfies G(x)= x*(1 + G(x)^2/2). - Geoffrey Critzer, Nov 13 2011
D-finite with recurrence (with no interpolated zeros): -2*a(n)*(n + 1)*(2*n + 1)*(2*n + 3) + (n + 2)*a(n+1) = 0 with a(0) = 1. - Petros Hadjicostas, Jun 08 2019
G.f.: 4F1(1,1,1/2,3/2;2;8*x). - R. J. Mathar, Jan 28 2020

A036772 Number of labeled rooted trees with a degree constraint: ((4n)!/(24^n)) binomial(4*n+1, n).

Original entry on oeis.org

1, 5, 2520, 9909900, 150089940000, 6217438242015000, 574985352122181000000, 103753754577643425255000000, 33189544956070738228953960000000, 17517292900368819935211385551000000000, 14427024664929016470240101675459976000000000

Offset: 0

N. J. A. Sloane

nonn

L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10; see Eq. (13) on p. 4 (with r = 4).
Index entries for sequences related to rooted trees

Cf. A036770, A036771, A036773.

Table[(4n)!/24^n Binomial[4n+1,n],{n,0,10}] (* Harvey P. Dale, Aug 10 2011 *)

From Petros Hadjicostas, Jun 08 2019: (Start)
Recurrence (with no interpolated zeros): -8 * (4*n + 1) * (4*n + 3)^2 * (2*n + 1)^2 * (4*n + 5) * a(n) + (81*n^2 + 162*n + 72) * a(n + 1) = 0 for n >= 0 with a(0) = 1.
E.g.f. (with interpolated zeros): Let G(x) = Sum_{n >= 0} a(n)*x^(4*n + 1)/(4*n + 1)!. Then the e.g.f. satisfies G(x) = x * (1 + G(x)^4/4!).
(End)

A036773 Number of labeled rooted trees with a degree constraint: ((5n)!/(120^n)) C(5*n+1, n).

Original entry on oeis.org

1, 6, 13860, 423783360, 70220478968640, 41004669682770393600, 65405789473547026656472320, 240729724316568938710767014707200, 1813083536072533851678174232377806438400, 25541737277107694920826740625991927645705830400

Offset: 0

N. J. A. Sloane

nonn

L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10; see Eq. (13) on p. 4 (with r = 5).
Index entries for sequences related to rooted trees

Cf. A036770, A036771, A036772.

E.g.f. with interpolated zeros: Let G(x) = Sum_{n >= 0} a(n)*x^(5*n + 1)/(5*n + 1)!. Then this e.g.f. satisfies the equation G(x) = x*(1 + G(x)^5/5!). - Petros Hadjicostas, Jun 08 2019

A367625 (A036991(n)-1)/2.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 115, 117, 118, 119, 121, 122, 123, 125, 126, 127

Offset: 2

N. J. A. Sloane, Nov 27 2023

nonn

N. J. A. Sloane, Table of n, a(n) for n = 2..13496

Cf. A036771, A367626, A367627.

from itertools import count, islice
def A367625_gen(startvalue=0): # generator of terms >= startvalue
    if startvalue <= 0:
        yield 0
    for n in count(max(startvalue,1)):
        s = bin(n)[2:]
        c, l = 2, len(s)
        for i in range(1,l+1):
            c += int(s[l-i])<<1
            if c <= i:
                break
        else:
            yield n
A367625_list = list(islice(A367625_gen(),30)) # Chai Wah Wu, Nov 28 2023

cp's OEIS Frontend

A036770 Number of labeled rooted trees with a degree constraint: (2n)!/(2^n) C(2*n+1, n).

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A036772 Number of labeled rooted trees with a degree constraint: ((4n)!/(24^n)) binomial(4*n+1, n).

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A036773 Number of labeled rooted trees with a degree constraint: ((5n)!/(120^n)) C(5*n+1, n).

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A367625 (A036991(n)-1)/2.

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cp's OEIS Frontend

A036770 Number of labeled rooted trees with a degree constraint: (2*n)!/(2^n) * C(2*n+1, n).

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Author

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Magma

Maple

Mathematica

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A036772 Number of labeled rooted trees with a degree constraint: ((4*n)!/(24^n)) * binomial(4*n+1, n).

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Links

Crossrefs

Programs

Mathematica

Formula

A036773 Number of labeled rooted trees with a degree constraint: ((5*n)!/(120^n)) * C(5*n+1, n).

Views

Author

Keywords

Links

Crossrefs

Formula

A367625 (A036991(n)-1)/2.

Views

Author

Keywords

Links

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Programs

Python

A036770 Number of labeled rooted trees with a degree constraint: (2n)!/(2^n) C(2*n+1, n).

A036772 Number of labeled rooted trees with a degree constraint: ((4n)!/(24^n)) binomial(4*n+1, n).

A036773 Number of labeled rooted trees with a degree constraint: ((5n)!/(120^n)) C(5*n+1, n).