A027610 -id:A027610 - cp's OEIS frontend

A047776 Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).

0, 0, 0, 0, 2, 11, 71, 370, 2005, 10682, 58167, 320116, 1789210, 10121965, 57933469, 334919626, 1953800059, 11489466014, 68053583772, 405713887061, 2433000197471, 14668527134167, 88869448492895, 540834097467624, 3304961431043989, 20273201718862728, 124798671079300720, 770762029389852807

Offset: 1

N. J. A. Sloane

One of 17 different symmetry types comprising A007173 and A027610 and one of 7 for A371350. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both asymmetric (type A) with n tetrahedral cells. The order of the symmetry group is 1. Each member of a chiral pair is a reflection but not a rotation of the other. - Robert A. Russell, Mar 31 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047776 examples

Cf. A007173 (oriented), A027610 (unoriented), A371350 (chiral), A001764 (rooted), A047775 (type B), A047774 (type C). A047773 (type D), A047762 (type E), A047760 (type F), A047758 (type G), A047754 (type H), A047753 (type I), A047752 (type J), A047751 (type K), A047771 (type L), A047769 (type M), A047766 (type N|O), A047765 (type P), A047764 (type Q).

Table[If[n < 5, 0, Binomial[3 n, 2 n + 2]/(3 n (n - 1))
    - If[OddQ[n], Binomial[3 n/2 - 1/2, n + 1] 3/(n - 1),
     7 Binomial[3 n/2, n + 1]/(3 n)]
    - Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 + 1/3]/(n - 1), 2,
     Binomial[n - 1, 2 n/3 + 2/3]/(n - 2), _, 0]
    + Switch[Mod[n, 4], 1, Binomial[3 n/4 - 3/4, n/2 + 1/2] 2/(3 (n - 1))
      + Binomial[3 n/4 + 1/4, n/2 + 3/2] 4/(n - 1) +
      Binomial[3 n/4 - 3/4, n/2 + 1/2] 4/(n + 3), 2,
     Binomial[3 n/4 - 1/2, n/2 + 1] 8/(n - 2), 3,
     Binomial[3 n/4 - 1/4, n/2 + 3/2] 12/(n - 3), 0,
     Binomial[3 n/4 - 1, n/2 + 1] 12/(n - 4)] +
    Switch[Mod[n, 6], 1, Binomial[n/2 - 1/2, n/3 + 2/3] 6/(n - 1), 2,
     Binomial[n/2 - 1, n/3 + 1/3] 4/(n - 2) +
      Binomial[n/2, n/3 + 4/3] 6/(n - 2) +
      Binomial[n/2 - 1, n/3 + 1/3] 6/(n + 4), 4,
     Binomial[n/2 - 1, n/3 + 2/3] 12/(n - 4), 5,
     Binomial[n/2 - 1/2, n/3 + 1/3] 9/(n + 4), _, 0] +
    Switch[Mod[n, 12], 2, -Binomial[n/4 - 1/2, n/6 + 2/3] 12/(n - 2), 5,
     Binomial[n/4 - 5/4, n/6 - 5/6] 2/(n + 1),
     8, -Binomial[n/4 - 1, n/6 - 1/3] 12/(n + 4), _, 0] -
    Switch[Mod[n, 24], 5, Binomial[n/8 - 5/8, n/12 - 5/12] 12/(n + 7), 17,
     Binomial[n/8 - 9/8, n/12 - 5/12] 24/(n + 7), , 0]]/2, {n, 1, 60}] (* _Robert A. Russell, Apr 09 2012 *)

From Robert A. Russell, Mar 31 2024: (Start)
a(n) = A001764(n)/(12(n+1)) - A047775(n)/2 - A047774(n)/3 - A047773(n)/6 - A047762(n)/2 - A047760(n)/4 - A047758(n)/4 - A047754(n)/4 - A047753(n)/8 - A047752(n)/12 - A047751(n)/24 - A047771(n)/2 - A047769(n)/2 - A047766(n)/6 - A047766(n)/6 - A047765(n)/4 - A047764(n)/12.
G.f.: (G(z^4) + G(z^6) - 2)/(2z) - z/3 + G(z)/6 - G(z)^2/12 + z*G(z)^4/24 - 7*G(z^2)/12 - 3z*G(z^2)^2/8 - z*G(z^3)/6 - z^2*G(z^3)^2/12 + G(z^4)/2 - z*G(z^4)/6 + (z*G(z^4)^2 + z^2*G(z^4)^2 + z*G(z^6))/2 + z^2*G(z^6)/12 + (z^2*G(z^6)^2 + z^4*G(z^6)^2 - z^2*G(z^12))/2 + z^5*G(z^12)/6 - (z^8*G(z^12)^2 + z^5*G(z^24) + z^17*G(z^24)^2)/2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

A001762 Number of labeled n-vertex dissections of a ball.

Original entry on oeis.org

1, 1, 10, 180, 4620, 152880, 6168960, 293025600, 15990004800, 984647664000, 67493121696000, 5094263446272000, 419688934689024000, 37465564582397952000, 3601861863990534144000, 370962724717928318976000, 40744403224500159055872000

Offset: 3

N. J. A. Sloane

nonn

This is the number of labeled Apollonian networks (planar 3-trees). - Allan Bickle, Feb 20 2024

			There is one maximal planar graph with 4 vertices, and one way to label it, so a(4) = 1.

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 3..100
L. W. Beineke and R. E. Pippert, The Number of Labeled Dissections of a k-Ball, Math. Annalen, 191 (1971), 87-98.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

Cf. A000407, A001763, A001764, A027610.

using Combinatorics
a(n) = n < 4 ? 1 : binomial(BigInt(n),3)*factorial(BigInt(3*n-9))÷factorial(BigInt(2*n-4))
print([a(n) for n in 3:28]) # Paul Muljadi, Mar 27 2024

Join[{1}, Table[Binomial[n, 3]*(3*n - 9)!/(2*n - 4)!, {n, 4, 25}]] (* T. D. Noe, Aug 10 2012 *)

from math import factorial
from sympy import binomial
def a(n):
    if n < 4:
        return 1
    else:
        return binomial(n, 3) * factorial(3*n-9) // factorial(2*n-4)
print([a(n) for n in range(3, 21)]) # Paul Muljadi, Mar 05 2024

a(n) = binomial(n,3)*(3*n-9)!/(2*n-4)!, n >= 4; a(3) = 1.

a(n) ~ 3^(3*n - 19/2) * n^(n-2) / (2^(2*n - 5/2) * exp(n)). - Vaclav Kotesovec, Mar 14 2024

More terms from Wolfdieter Lang

Name clarified by Andrey Zabolotskiy, Mar 15 2024

A121180 Alkane systems (see Cyvin reference for precise definition).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 26, 32, 133, 176, 708, 952, 3861, 5302, 21604, 29960

Offset: 1

N. J. A. Sloane, Aug 17 2006

Appears to be A047774 without every third term (all omitted terms are zeros). - Andrey Zabolotskiy, Jul 29 2023

S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239. See Table 5, column C_2.

Cf. other columns of Cyvin et al.'s Table 5: A027610 (spectral isomers), A007173 (stereoisomers), A047775 (C_s), A047772 (C_i), A047774 (C_3, apparently), A047767 (C_{2h}), A047761 (C_{2v}), A047773 (C_{3v}, apparently).

Cf. A121181, A121189.

cp's OEIS Frontend

A047776 Number of chiral pairs of asymmetric dissectable polyhedra with n tetrahedral cells (type A).

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A001762 Number of labeled n-vertex dissections of a ball.

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A121180 Alkane systems (see Cyvin reference for precise definition).

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