A000625 -id:A000625 - cp's OEIS frontend

A005960 Number of acyclic disubstituted alkanes with n carbon atoms and distinct substituents.

1, 3, 8, 23, 69, 208, 636, 1963, 6099, 19059, 59836, 188576, 596252, 1890548, 6008908, 19139155, 61074583, 195217253, 624913284, 2003090071, 6428430129, 20653101216, 66420162952, 213802390264, 688796847976, 2220789746752, 7165331870036

Offset: 1

N. J. A. Sloane

nonn

Acyclic alkanes of the form C_nH_mXY with X != Y. For X = Y see A005961. - Sean A. Irvine, Nov 12 2016

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.

Cf. A193418.

terms = 27; (* B = g.f. for A000625 *) B[] = 0; Do[B[x] = 1 + x*((B[x]^3 + 2*B[x^3])/3) + O[x]^(terms + 1) // Normal, terms + 1];
A[x_] = 1/(1 - x*B[x]^2) + O[x]^(terms + 1);
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 10 2018 *)

G.f.: 1/(1-x*B(x)^2) where B(x) is the g.f. for A000625. - Sean A. Irvine, Nov 12 2016

More terms and title improved by Sean A. Irvine, Nov 12 2016

A287211 The number of plane rooted complete ternary trees with 2n+1 unlabeled leaves (hence n internal nodes including the root where n starts at 0) satisfying these two conditions: (1) if one of the three children of any internal node is the greatest in deglex order then that child is not the leftmost child; (2) if one of the three children of any internal node is the smallest in deglex order then that child is not the rightmost child. Deglex order refers to degree-lexicographical order defined inductively on the number of leaves (see details under Comments).

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 308, 1264, 5332, 22994, 100896, 449004

Offset: 0

Murray R. Bremner, May 21 2017

"Plane" means "embedded in the plane" or (equivalently) the three children of each internal node (including the root) are ordered left, middle, right. Deglex order on trees with 2n+1 leaves is defined as follows: to compare two such trees T and U with children T_1, T_2, T_3 and U_1, U_2, U_3, first find the least index 1 <= i <= 3 for T_i <> U_i, then compare T_i and U_i in deglex order already defined inductively on trees with fewer than 2n+1 leaves; note that this requires comparing trees with different numbers of leaves, so we say that T_i precedes U_i if either (i) T_i has fewer leaves than U_i, or (ii) T_i and U_i have the same number of leaves, and T_i precedes U_i in deglex order.
An alternative description of this sequence: it counts the distinct association types in arity 2n+1 for a ternary operation [a,b,c] satisfying the cyclic-sum relation [a,b,c] + [b,c,a] + [c,a,b] = 0. The two conditions stated under "Name" are necessary to deal with the possibility of repeated factors: [a,a,b], [a,b,a], [b,a,a] where a < b in deglex order, and [a,b,b], [b,a,b], [b,b,a] where a < b in deglex order.
See further details in the comments to the Maple program which is attached as a a-file.

			Association types for arities 1, 3, 5, 7 are as follows in deglex order. See Links for a-file with association types for arities up to 11.
Arity 1, number of types 1:
a.
Arity 3, number of types 1:
[abc].
Arity 5, number of types 2:
[ab[cde]],
[a[bcd]e].
Arity 7, number of types 6:
[ab[cd[efg]]],
[ab[c[def]g]],
[a[bcd][efg]],
[a[bc[def]]g],
[a[b[cde]f]g],
[[abc]d[efg]].

Murray R. Bremner, Association types up to and including arity 11
Murray R. Bremner, Maple worksheet for generating association types

Cf. A357538, A000598, A000625, A001764, A006629, A036370, A036437, A233389.

Maple
```
See attached a-file under Links.
```

A375438 Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 12, 24, 46, 93, 191, 393, 819, 1724, 3648, 7772, 16654, 35850, 77520, 168289, 366629, 801328, 1756620, 3861157, 8508247, 18791480, 41591566, 92237598, 204931918, 456096480, 1016720162, 2269865456, 5074732028, 11360680664, 25464831258, 57146836290

Offset: 1

Paul D. Hanna, Aug 21 2024

nonn

			G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 24*x^9 + 46*x^10 + 93*x^11 + 191*x^12 + 393*x^13 + 819*x^14 + 1724*x^15 + ...
where A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 36*x^8 + 70*x^9 + 138*x^10 + 279*x^11 + 571*x^12 + 1179*x^13 + 2457*x^14 + 5168*x^15 + ...
Let B(x) be the series reversion, B(A(x)) = x, then B(x) begins
B(x) = x - x^2 + x^3 - x^4 + 3*x^6 - 8*x^7 + 9*x^8 + 18*x^9 - 134*x^10 + 442*x^11 - 997*x^12 + 1428*x^13 - 10*x^14 - 7640*x^15 + ...
SPECIFIC VALUES.
A(2/5) = 0.741461459188681119672668058998130332678610537393868...
A(1/3) = 0.515838591521774544528452689654484632143493145820237...
A(1/4) = 0.335691575266570204286454430830296229544471248787335...
A(1/5) = 0.250623759777806277498640241328479184446996870792060...
A(1/6) = 0.200220725056085320106333620370114891484630832364434...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A375439, A357538, A000625.

{a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);
A[#A] = polcoeff( x + x^2 + ( Ax^3 + 2*subst(Ax,x,x^3))/3 - Ax,#A-1) );A[n+1]}
for(n=1,40,print1(a(n),", "))

a(n) ~ c * d^n / n^(3/2), where d = 2.3394011548205191342723840863090815005163727107... and c = 0.278795592719328257510209001410993009380027455... - Vaclav Kotesovec, Aug 22 2024

A005628 Number of chiral planted trees with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 20, 60, 176, 510, 1484, 4314, 12624, 37126, 109864, 326958, 978528, 2943384, 8895792, 27001378, 82281216, 251636434, 772101086, 2376186784, 7333094178, 22688117658, 70360646672, 218678194238, 681016789056

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000625, A005627.

s[0]:=1:s[1]:=1:for n from 0 to 60 do s[n+1/3]:=0 od:for n from 0 to 60 do s[n+2/3]:=0 od:for n from 1 to 55 do s[n+1]:=(2*n/3*s[n/3]+sum(j*s[j]*sum(s[k]*s[n-j-k],k=0..n-j),j=1..n))/n od:p[0]:=1: for n from 0 to 50 do p[n+1]:=sum(s[k]*p[n-2*k],k=0..floor(n/2)) od:seq(s[n]-p[n],n=0..37); # here s[n]=A000625 and p[n]=A005627(n)

nmax = 28;
s[0] = s[1] = 1; s[_] = 0;
Do[s[n+1] = (2*n/3*s[n/3] + Sum[j*s[j]*Sum[s[k]*s[n-j-k], {k, 0, n-j}], {j, 1, n}])/n, {n, 1, nmax}];
p[0] = 1;
Do[p[n+1] = Sum[s[k]*p[n-2*k], {k, 0, Floor[n/2]}], {n, 0, nmax}];
a[n_] := s[n] - p[n];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 07 2024, after Maple code *)

a(n) = A000625(n)-A005627(n) (given as g(n)=s(n)-p(n) on p. 357 of the Robinson et al. paper). - Emeric Deutsch, May 16 2004

More terms from Emeric Deutsch, May 16 2004

A005629 Number of achiral trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 14, 21, 40, 61, 118, 186, 355, 567, 1081, 1755, 3325, 5454, 10306, 17070, 32136, 53628, 100704, 169175, 316874, 535267, 1000524, 1698322, 3168500, 5400908, 10059823, 17211368, 32010736, 54947147, 102059572, 175702378

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Bytautats and D. J. Klein, Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803, Table 1.
R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000625, A005627.

s[0]:=1:s[1]:=1:for n from 0 to 60 do s[n+1/3]:=0 od:for n from 0 to 60 do s[n+2/3]:=0 od:for n from 1 to 55 do s[n+1]:=(2*n/3*s[n/3]+sum(j*s[j]*sum(s[k]*s[n-j-k],k=0..n-j),j=1..n))/n od: p[0]:=1: for n from 0 to 50 do > p[n+1]:=sum(s[k]*p[n-2*k],k=0..floor(n/2)) od:seq(p[j],j=0..45): P:=proc(n) if floor(n)=n then p[n] else 0 fi end:S:=proc(n) if floor(n)=n then s[n] else 0 fi end:t:=n->(P(n)+S(n/2)+S((n-1)/4))/2: seq(t(n),n=1..40); # here s[n]=A000625(n), p[n]=A005627(n). - Emeric Deutsch, Nov 21 2004

nmax = 37;
s[0] = s[1] = 1; s[_] = 0;
Do[s[n + 1] = (2*n/3*s[n/3] + Sum[j*s[j]*Sum[s[k]*s[n - j - k], {k, 0, n - j}], {j, 1, n}])/n, {n, 1, nmax}];
p[0] = 1;
Do[p[n + 1] = Sum[s[k]*p[n - 2 k], {k, 0, Floor[n/2]}]; a[n + 1] = (p[n + 1] + s[(n + 1)/2] + s[n/4])/2, {n, 0, nmax}];
a[n_] := s[n] - p[n];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Jul 07 2024, after Maple code *)

a(n+1) = (p(n+1)+s((n+1)/2)+s(n/4))/2, where p(n)=A005627(n) and s(n)=A000625(n) (eq. (23) in the Robinson et al. reference). - Emeric Deutsch, Nov 21 2004

Corrected and extended by Emeric Deutsch, Nov 21 2004

A005630 Number of chiral trees with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 10, 34, 96, 284, 782, 2226, 6208, 17560, 49618, 141500, 405104, 1168316, 3386538, 9875232, 28939944, 85235762, 252159572, 749160544, 2234378156, 6688358338, 20088296030, 60524845158, 182893018756

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Bytautats and D. J. Klein, Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803, Table 1.
R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Index entries for sequences related to trees

a(n) = A000628(n) - A005629(n). - Emeric Deutsch, Nov 21 2004

Corrected and extended by Emeric Deutsch, Nov 21 2004

A005957 Number of acyclic ketone and aldehyde stereo-isomers with n carbon atoms.

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 47, 123, 338, 935, 2657, 7616, 22138, 64886, 191873, 571169, 1711189, 5153883, 15599094, 47415931, 144692886, 443091572, 1361233280, 4194107380, 12957209782, 40128629400, 124562327944, 387467513732, 1207633413980

Offset: 1

N. J. A. Sloane

nonn

It seems that, starting with a(2), this is row j = 1 in Table 3 (p. 176) in Wang et al. (2003). This is also corroborated by the title of the table: "The numbers f_{ij} of stereo isomers of acyclic saturated compounds consisting of C, O, and H, C_i H_{2i+2} O_j". Such a claim, if true, can probably be proved by using Eqs. (5), (6), and (7) on pp. 175-176 of the paper. - Petros Hadjicostas, Jul 24 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.
Jianji Wang, Ruxiong Li, and Shen Wang, Enumeration of isomers of acyclic saturated hydroxyl ethers, Journal of Mathematical Chemistry 33 (2003), 171-179.

G.f.: (1/2)*x*(B^2(x)+B(x^2)) where B(x) is the g.f. for A000625. - Sean A. Irvine, Nov 12 2016

a(23) corrected, title improved, and more terms from Sean A. Irvine, Nov 12 2016

A005961 Number of acyclic disubstituted alkanes with n carbon atoms and identical substituents.

Original entry on oeis.org

1, 2, 5, 13, 37, 108, 325, 993, 3070, 9564, 29979, 94392, 298311, 945592, 3005021, 9570559, 30539044, 97611676, 312462096, 1001554565, 3214232129, 10326580526, 33210135104, 106901289420, 344398593149, 1110395171502, 3582666470285, 11567123373968

Offset: 1

N. J. A. Sloane

nonn

Acyclic alkanes of the form C_nH_mX2. - Sean A. Irvine, Nov 12 2016

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.

Cf. A005960.

G.f.: -1 + (1/(1-x*B(x)^2) + 1/(1-x*B(x^2)))/2 where B(x) is the g.f. for A000625. - Sean A. Irvine, Nov 12 2016

More terms and title improved by Sean A. Irvine, Nov 12 2016

A005955 Number of acyclic secondary alcohols with n carbon atoms.

Original entry on oeis.org

1, 2, 5, 14, 36, 98, 273, 768, 2197, 6360, 18584, 54780, 162672, 486154, 1461197, 4413988, 13393855, 40807290, 124783669, 382842018, 1178140280, 3635626680, 11247841224, 34880346840, 108402132544, 337576497920, 1053229358252, 3291813720292

Offset: 3

N. J. A. Sloane

nonn

			G.f. = x^3 + 2*x^4 + 5*x^5 + 14*x^6 + 36*x^7 + 98*x^8 + 273*x^9 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.

Cf. A000625.

G.f. x*(B(x)-1)^2 where B(x) is the g.f. for A000625. - Sean A. Irvine, Nov 12 2016

More terms from Sean A. Irvine, Nov 12 2016

A005956 Number of acyclic tertiary alcohols with n carbon atoms.

Original entry on oeis.org

1, 1, 3, 10, 27, 79, 234, 686, 2036, 6080, 18224, 54920, 166245, 505201, 1541014, 4716540, 14480699, 44586619, 137648341, 425992838, 1321362034, 4107332002, 12792440104, 39915889304, 124762916912, 390594099694, 1224682772872, 3845389399882

Offset: 4

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.

G.f. (1/3)*x*((B(x)-1)^3 + 2*(B(x^3)-1)) where B(x) is the g.f. for A000625. - Sean A. Irvine, Nov 12 2016

More terms from Sean A. Irvine, Nov 12 2016

cp's OEIS Frontend

A005960 Number of acyclic disubstituted alkanes with n carbon atoms and distinct substituents.

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A375438 Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (A(x)^3 + 2*A(x^3))/3.

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A005628 Number of chiral planted trees with n nodes.

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A005629 Number of achiral trees with n nodes.

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A005630 Number of chiral trees with n nodes.

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A005957 Number of acyclic ketone and aldehyde stereo-isomers with n carbon atoms.

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A005961 Number of acyclic disubstituted alkanes with n carbon atoms and identical substituents.

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