A000625 -id:A000625 - cp's OEIS frontend

A000622 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

0, 0, 0, 2, 4, 14, 34, 98, 270, 768, 2192, 6360, 18576, 54780, 162658, 486154, 1461174, 4413988, 13393816, 40807290, 124783604, 382842018, 1178140170, 3635626680, 11247841040, 34880346840, 108402132234, 337576497920, 1053229357732, 3291813720292, 10305275270364

Offset: 1

N. J. A. Sloane

nonn

R-CH-X (secondary)
.....|
.....R'
Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = this sequence, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1930
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000620-A000625, A239803, A239807.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.171310881484463744523... (see A239807). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A000623 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 18, 66, 208, 646, 1962, 5962, 18014, 54578, 165650, 504220, 1539330, 4713742, 14475936, 44578668, 137634872, 425970290, 1321323952, 4107268140, 12792332438, 39915708564, 124762612530, 390593588402, 1224681912368, 3845387953884, 12090382743374

Offset: 1

N. J. A. Sloane

nonn

....X
....|
R-C-R' (tertiary)
....|
....R"
Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = this sequence, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1930
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000620-A000625, A239803, A239808.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.069641252627300354415... (see A239808). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A000624 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are not stereoisomers.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 9, 13, 26, 40, 74, 118, 210, 342, 595, 981, 1684, 2798, 4763, 7951, 13469, 22548, 38082, 63862, 107666, 180740, 304382, 511292, 860504, 1445998, 2432665, 4088805, 6877172, 11560684, 19441791, 32684789, 54961955, 92404472, 155377371, 261235027

Offset: 1

N. J. A. Sloane

nonn

....X
....|
R-C-R' (tertiary)
....|
....R"
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = this sequence, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..3000
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000620-A000625, A239804, A239809.

a(n) ~ c * beta^n, where beta = 1.681367524441880255591... (see A239804), c = 0.146177958025494272954... (see A239809). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

False g.f. deleted by N. J. A. Sloane, May 13 2008

A002094 Number of unlabeled connected loop-less graphs on n nodes containing exactly one cycle (of length at least 2) and with all nodes of degree <= 4.

Original entry on oeis.org

0, 1, 2, 5, 10, 25, 56, 139, 338, 852, 2145, 5513, 14196, 36962, 96641, 254279, 671640, 1781840, 4742295, 12662282, 33898923, 90981264, 244720490, 659591378, 1781048728, 4817420360, 13050525328, 35405239155, 96180222540, 261603173201, 712364210543

Offset: 1

N. J. A. Sloane

nonn

A pair of parallel edges is permitted and is regarded as a cycle of length 2.
The original definition in A Handbook of Integer Sequences (1973) based on Schiff (1875) was simply "Alcohols". - N. J. A. Sloane, Mar 22 2018
Schiff used an now outdated terminology and did not use the term "alcohols", but in German "zweiwerthige Kohlenwasserstoffe C_{n}H_{2n} ..." and later "... deren je zwei verfuegbare Affinitaeten ... durch Alkoholradikale befriedigt sind.", translated "bivalent hydrocarbons ... whose free valences ... are covered by alcohol radicals". At that time the meaning of "alcohol radical" was different from modern terminology, now meaning an -OH group, but in Schiff's terminology another C_{x}H{y} hydrocarbon group was meant. The organic compounds that are described by the graphs of this sequence in modern chemical terminology are the acyclic alkenes, with exactly one double carbon-to-carbon bond, and the monocyclic cycloalkanes (see Wikipedia links). - Hugo Pfoertner, Mar 29 2018

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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..400
R. J. Mathar, Illustration for graphs up to 6 carbons, 2018
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875.
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
Wikipedia, Alkene. Those with exactly one double carbon-to-carbon bond are covered by this sequence, the simplest being ethylene C_{2}H_{4}.
Wikipedia, Cycloalkane. The simplest alicyclic compounds, which are the monocyclic saturated hydrocarbons with formula C_{n}H_{2n}, are covered by this sequence, the first example being cyclopropane C_{3}H_{6}.

Cf. A000294, A000598, A000602, A000625, A000642, A001429 (unbound degrees), A068051.

# cycle index of cyclic group C_n
cycC_n := proc(n::integer,a)
    local d ;
    add(numtheory[phi](d)*a[d]^(n/d),d=numtheory[divisors](n)) ;
    %/n ;
end proc:
# cycle index of dihedral group
cyD_n := proc(n::integer,a)
    cycC_n(n,a)/2 ;
    if type(n,'odd') then
        %+ a[1]*a[2]^((n-1)/2)/2 ;
    else
        %+ ( a[1]^2*a[2]^((n-2)/2) +a[2]^(n/2) )/4 ;
    end if;
end proc:
a000642 := [
    1, 1, 2, 3, 7, 14, 32, 72, 171, 405, 989, 2426, 6045, 15167, 38422, 97925,
    251275, 648061, 1679869, 4372872, 11428365, 29972078, 78859809, 208094977,
    550603722, 1460457242, 3882682803, 10344102122, 27612603765, 73844151259,
    197818389539, 530775701520, 1426284383289] ;
g := [add(a000642[i]*x^i,i=1..nops(a000642)) ];
for i from 2 to nops(a000642) do
    g := [op(g), subs(x=x^i,g[1]) ] ;
end do:
Nmax := nops(a000642) :
G := 0 ;
for c from 2 to Nmax do
    f := cyD_n(c,g) ;
    G := G+ taylor(f,x=0,Nmax) ;
end do:
taylor(G,x=0,Nmax) ;
gfun[seriestolist](%) ; # R. J. Mathar, Mar 17 2018

terms = 31;
cycC[n_, a_] := Sum[EulerPhi[d] a[[d]]^(n/d), {d, Divisors[n]}]/n;
cyD[n_, a_] := Module[{cc}, cc = (1/2)cycC[n, a]; If[OddQ[n], (1/2)a[[1]]* a[[2]]^((n-1)/2)+cc, (1/4)(a[[1]]^2 a[[2]]^((n-2)/2) + a[[2]]^(n/2)) + cc]];
B[] = 0; Do[B[x] = Normal[(1/6) x (2 B[x^3] + 3 B[x^2] B[x] + B[x]^3) + O[x]^terms+1], terms];
A[x_] = (1/2) x (B[x^2] + B[x]^2) + O[x]^(terms+2);
a000642 = Rest[CoefficientList[A[x], x]];
g = {Sum[x^i a000642[[i]], {i, 1, terms+1}]};
For[i = 2, i <= Length[a000642], i++, g = Flatten[Append[g, g[[1]] /. x -> x^i]]];
For[G = 0; c = 2, c < terms+1, c++, f = cyD[c, g]; G = Series[f, {x, 0, terms+1}] + G];
Most[Rest[CoefficientList[G, x]]] (* Jean-François Alcover, Mar 26 2020, after R. J. Mathar *)

Let A(x) denote the generating function for A000598 (Number of rooted ternary trees with n nodes), i.e., A(x) = 1+(1/6)*x*(A(x)^3+3*A(x)*A(x^2)+2*A(x^3)), and set B(x)=(A(x)^2+A(x^2))/2. With D_k(x) being the cycle polynomial of the regular k-gon for k>=2, the final generating function is then given by Sum_{k>=2} x^k*D_k(B(x)), which can be evaluated very quickly. - Sascha Kurz, Mar 18 2018

Better definition from R. J. Mathar; terms beyond 852 from R. J. Mathar and Sean A. Irvine, Mar 17 2018

A086194 Number of unrooted steric quartic trees with n (unlabeled) nodes and possessing a centroid; number of n carbon alkanes C(n)H(2n +2) with a centroid when stereoisomers are regarded as different.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 11, 9, 55, 70, 345, 494, 2412, 3788, 18127, 30799, 143255, 256353, 1173770, 2190163, 9892302, 19130814, 85289390, 169923748, 749329719, 1531701274, 6688893605, 13984116304, 60526543480, 129073842978

Offset: 1

Steve Strand (snstrand(AT)comcast.net), Aug 28 2003

nonn

The degree of each node is <= 4.
A centroid is a node with less than n/2 nodes in each of the incident subtrees, where n is the number of nodes in the tree. If a centroid exists it is unique.
Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A010372 for the analogous sequence when stereoisomers are not counted as different.

Cf. A000598, A000602, A010732, A010733, A000625, A000628, A086200.

For even n A000628(n) = a(n) + A086200(n/2), for odd n A000628(n) = a(n), since every tree has either a centroid or a bicentroid but not both.

c[0] = 1; f[x_, m_] := Sum[c[k] x^k, {k, 0, m}]; coes[m_] := CoefficientList[Series[f[x, m] - 1 - (x*(f[x, m]^3 + 2*f[x^3, m])/3), {x, 0, m}], x] // Rest; r[x_, m_] := r[x, m] = (f[x, m] /. Solve[Thread[coes[m] == 0]] // First); b[m_] := CoefficientList[(1/12)*(r[x, m]^4 + 3*r[x^2, m]^2 + 8*r[x, m]*r[x^3, m]), x]; a[1]=1; a[2]=0; a[n_] := b[Quotient[n-1, 2]][[n]]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 30}] (* Jean-François Alcover, Dec 29 2014 *)

Let r(x) = g.f. A(x) for A000625 truncated after the x^n term (x^0 through x^n terms only). Then coefficients of x^(2n) and x^(2n+1) in [r(x)^4 + 8 r(x^3) r(x) + 3 r(x^2)^2]/12 are terms 2n+1 and 2n+2 in current sequence..

A000632 Number of esters with n carbon atoms up to structural isomerism.

Original entry on oeis.org

1, 2, 4, 9, 20, 45, 105, 249, 599, 1463, 3614, 9016, 22695, 57564, 146985, 377555, 974924, 2529308, 6589734, 17234114, 45228343, 119069228, 314368027, 832193902, 2208347917, 5873364623, 15653499416, 41800070483, 111821751649

Offset: 2

N. J. A. Sloane

nonn

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
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).

Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008, Table of n, a(n) for n = 2..99
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
H. R. Henze and C. M. Blair, The number of structural isomers of the more important types of aliphatic compounds, J. Amer. Chem. Soc., 56 (1) (1934), 157-157.
H. R. Henze and C. M. Blair, The number of structural isomers of the more important types of aliphatic compounds, J. Amer. Chem. Soc., 56 (1) (1934), 157-157. (Annotated scanned copy)
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. 28.
Wikipedia, Ester.

terms = 29; (* B = g.f. for A000625 *) B[] = 0; Do[B[x] = 1 + (1/6)*x*(B[x]^3 + 3*B[x]*B[x^2] + 2*B[x^3]) + O[x]^(terms+2) // Normal, terms+2];
A[x_] = 1*x*B[x]*(B[x] - 1) + O[x]^(terms+2);
Drop[CoefficientList[A[x], x], 2] (* Jean-François Alcover, Jan 10 2018 *)

G.f.: A(x)=x*B(x)*(B(x)-1), where B(x) = g.f. for A000598. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

Name clarified by Sean A. Irvine, Feb 01 2025

A086200 Number of unrooted steric quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n +2) with a bicentroid when stereoisomers are regarded as different.

Original entry on oeis.org

1, 3, 15, 66, 406, 2775, 19900, 152076, 1206681, 9841266, 82336528, 702993756, 6105180250, 53822344278, 480681790786, 4342078862605, 39621836138886, 364831810979041, 3386667673687950, 31669036266203766

Offset: 1

Steve Strand (snstrand(AT)comcast.net), Aug 28 2003

nonn

The degree of each node is <= 4.
A bicentroid is an edge which connects two subtrees of exactly m/2 nodes, where m is the number of nodes in the tree. If a bicentroid exists it is unique. Clearly trees with an odd number of nodes cannot have a bicentroid.
Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A010373 for the analogous sequence when stereoisomers are not counted as different.

Cf. A000598, A000602, A010732, A010733, A000625, A000628, A086194.

For even n A000628(n) = A086194(n) + a(n/2), for odd n A000628(n) = A086194(n), since every tree has either a centroid or a bicentroid but not both.

G.f.: replace each term x in g.f. for A000625 by x(x+1)/2. Interpretation: ways to pick 2 specific radicals (order not important) from all n carbon radicals is number of 2n carbon bicentered alkanes (join the two radicals with an edge).

A357538 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + x(2A(x)^3 + A(x^3))/3.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 308, 1264, 5332, 22994, 100896, 449004, 2021712, 9193509, 42161222, 194768936, 905522052, 4233712140, 19893553120, 93894821200, 444952447944, 2116220266360, 10098086643002, 48330679370584, 231954451580616, 1116046254269592, 5382402925982248

Offset: 0

Paul D. Hanna, Dec 02 2022

nonn

Is this sequence the same as A287211?

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 78*x^5 + 308*x^6 + 1264*x^7 + 5332*x^8 + 22994*x^9 + 100896*x^10 + ...
where A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 117*x^4 + 462*x^5 + 1895*x^6 + 7998*x^7 + 34491*x^8 + 151341*x^9 + 673506*x^10 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A287211, A000625, A375439.

A357538 := proc(n)
    option remember ;
    if n < 0 then
        0;
    elif n <= 1 then
        1;
    else
        a := 0 ;
        for j from 0 to n-1 do
            a := a + procname(n-1-j)*add(procname(i)*procname(j-i),i=0..j)
        end do:
        a := 2*a/3 ;
        if modp(n-1,3) = 0 then
            a := a+procname((n-1)/3)/3 ;
        end if;
        a ;
    end if ;
end proc:
seq(A357538(n),n=0..20) ; # R. J. Mathar, Dec 19 2022

{a(n) = my(A=1); for(i=1,n, A = 1 + x*(2*A^3 + subst(A,x,x^3))/3 +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = if(n, my(A=vector(n+1)); A[1]=1; A[2]=1; for(k=1, n-1, A[k+2] = sum(j=1, k, 2*j*A[j+1]*(sum(i=0, k-j, A[i+1]*A[k-j-i+1])))/k + (1/3)*if(k%3, 0, A[k/3+1])); A[n+1], 1)} \\ after Jianing Song in A000625
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.

a(0) = a(1) = 1; a(n+1) = a(n/3)/3 + 2*(Sum_{j=0..n} a(n-j)*(Sum_{i=0..j} a(i)*a(j-i)))/3 for n >= 1, where a(k) = 0 if k not an integer (see formula and comment by Emeric Deutsch in A000625). (corrected by R. J. Mathar, Dec 19 2022)

A375439 Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 117, 290, 734, 1880, 4868, 12730, 33556, 89072, 237904, 638873, 1723930, 4672008, 12710904, 34703894, 95054188, 261116816, 719223064, 1985934212, 5496123033, 15242821108, 42357113994, 117918233704, 328833828334, 918470764376, 2569238134248, 7197046596440

Offset: 1

Paul D. Hanna, Aug 21 2024

nonn

Conjecture: a(n) is odd iff n is in A038754, which consists of numbers of the form 3^k and 2*3^k.

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 20*x^7 + 48*x^8 + 117*x^9 + 290*x^10 + 734*x^11 + 1880*x^12 + 4868*x^13 + 12730*x^14 + 33556*x^15 + ...
where A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 30*x^7 + 72*x^8 + 175*x^9 + 435*x^10 + 1101*x^11 + 2819*x^12 + 7302*x^13 + 19095*x^14 + 50332*x^15 + ...
Let B(x) be the series reversion of A(x), B(A(x)) = x, then B(x) begins
B(x) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 9*x^6 + 22*x^7 - 57*x^8 + 152*x^9 - 411*x^10 + 1119*x^11 - 3063*x^12 + 8436*x^13 - 23405*x^14 + 65452*x^15 + ...
SPECIFIC VALUES.
A(1/3) = 0.6046115975458048490061476622502250915528261368314569825...
where A(1/3) = 4/9 + (2*A(1/3)^3 + A(1/27))/3.
A(1/4) = 0.345218924086872316546119663994502755734706567000751...
A(1/5) = 0.253555647303827972834265469178971877524548605418192...
A(1/6) = 0.201444567662949882659512632012060178593075505771758...
A(1/7) = 0.167365364255434800795732539120237367470157092655512...
A(1/8) = 0.143236474390624253781858259379882809014038308155736...
A(1/27) = 0.03846365186207481603806452459437536518999937182129...

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

Cf. A375438, 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 + ( 2*Ax^3 + 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 = 2.9308423020191987018531615662206918839933116797613922... and c = 0.186346847470275688362452238277535367815900456286173... - Vaclav Kotesovec, Aug 22 2024

A005627 Number of achiral planted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 14, 23, 41, 69, 122, 208, 370, 636, 1134, 1963, 3505, 6099, 10908, 19059, 34129, 59836, 107256, 188576, 338322, 596252, 1070534, 1890548, 3396570, 6008908, 10801816, 19139155, 34422537, 61074583, 109894294, 195217253

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.

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:a[0]:=1: for n from 0 to 50 do a[n+1]:=sum(s[k]*a[n-2*k],k=0..floor(n/2)) od:seq(a[j],j=0..45); # here s[n]=A000625(n).

nmax = 36;
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}];
a[0] = a[1] = 1;
Do[a[n+1] = Sum[s[k]*a[n-2*k], {k, 0, Floor[n/2]}], {n, 1, nmax}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 07 2024, after Maple code *)

a(0)=1, a(n+1):=sum(s(k)*a(n-2*k), k=0..floor(n/2)) (n>=0), where s(n)=A000625(n) (this is eq. (15) in the Robinson et al. paper). - Emeric Deutsch, May 16 2004

More terms from Emeric Deutsch, May 16 2004

cp's OEIS Frontend

A000622 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

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A000623 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

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A000624 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are not stereoisomers.

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A002094 Number of unlabeled connected loop-less graphs on n nodes containing exactly one cycle (of length at least 2) and with all nodes of degree <= 4.

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A086194 Number of unrooted steric quartic trees with n (unlabeled) nodes and possessing a centroid; number of n carbon alkanes C(n)H(2n +2) with a centroid when stereoisomers are regarded as different.

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A000632 Number of esters with n carbon atoms up to structural isomerism.

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A086200 Number of unrooted steric quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n +2) with a bicentroid when stereoisomers are regarded as different.

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A357538 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + x*(2*A(x)^3 + A(x^3))/3.

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A357538 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + x(2A(x)^3 + A(x^3))/3.