A010733 -id:A010733 - cp's OEIS frontend

A000625 Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account.

1, 1, 1, 2, 5, 11, 28, 74, 199, 551, 1553, 4436, 12832, 37496, 110500, 328092, 980491, 2946889, 8901891, 27012286, 82300275, 251670563, 772160922, 2376294040, 7333282754, 22688455980, 70361242924, 218679264772, 681018679604, 2124842137550, 6641338630714, 20792003301836

Offset: 0

N. J. A. Sloane

Nodes are unlabeled, each node has out-degree <= 3.
Steric, or including stereoisomers, means that the children of a node are taken in a certain cyclic order. If the children are rotated it is still the same tree, but any other permutation yields a different tree. See A000598 for the analogous sequence with stereoisomers not counted.
Other descriptions of this sequence: steric planted trees with n nodes; total number of monosubstituted alkanes C(n)H(2n+1)-X with n carbon atoms.
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 = A000623, Tn = A000624, T = this sequence. Recurrences generating these sequences are given in the Maple program in A000620.

J. K. Percus, Combinatorial Methods, Lecture Notes, 1967-1968, Courant Institute, New York University, 212pp. See pp. 64-65.
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 = 0..1930 (first 200 terms from T. D. Noe)
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)
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, Eq. (25).
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, Eq. (25). (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. 44.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (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. A000598, A000602, A000620-A000624, A000628, A010732, A010733, A086194, A086200, A239803, A239810.

A := 1; f := proc(n) global A; coeff(series( 1+(1/3)*x*(A^3+2*subs(x=x^3,A)), x, n+1), x, n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;
A000625 := proc(n)
    local j,i,a ;
    option remember;
    if n <= 1 then
        1 ;
    else
        a :=0 ;
        for j from 1 to n-1 do
            a := a+ j*procname(j)*add(procname(i)*procname(n-j-i-1),i=0..n-j-1) ;
        end do:
        if modp(n-1,3) = 0 then
            a := a+2*(n-1)*procname((n-1)/3)/3 ;
        end if;
        a/ (n-1) ;
    end if;
end proc:
seq(A000625(n),n=0..30) ;

m = 31; c[0] = 1; gf[x_] = Sum[c[k] x^k, {k, 0, m}]; cc = Array[c, m]; coes = CoefficientList[ Series[gf[x] - 1 - (x*(gf[x]^3 + 2*gf[x^3])/3), {x, 0, m}], x] // Rest; Prepend[cc /. Solve[ Thread[ coes == 0], cc][[1]], 1]
(* Jean-François Alcover, Jun 24 2011 *)
a[0] = a[1] = 1; a[n_Integer] := a[n] = (Sum[j*a[j]*Sum[a[i]*a[n-i-j-1], {i, 0, n-j-1}], {j, 1, n-1}] + (2/3)*(n-1)*a[(n-1)/3])/(n-1); a[] = 0; Table[a[n], {n, 0, 31}] (* _Jean-François Alcover, Apr 21 2016, after Emeric Deutsch *)
terms = 32; A[] = 0; Do[A[x] = Normal[1 + x*(A[x]^3 + 2*A[x^3])/3 + O[x]^terms], terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Apr 22 2016, updated Jan 11 2018 *)

a(n) = if(n, my(v=vector(n+1)); v[1]=1; v[2]=1; for(k=1, n-1, v[k+2] = sum(j=1, k, j*v[j+1]*(sum(i=0, k-j, v[i+1]*v[k-j-i+1])))/k + (2/3)*if(k%3, 0, v[k/3+1])); v[n+1], 1) \\ Jianing Song, Feb 17 2019

G.f. A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 28*x^6 + ... satisfies A(x) = 1 + x*(A(x)^3 + 2*A(x^3))/3.
a(0) = a(1) = 1; a(n+1) = 2*a(n/3)/3 + (Sum_{j=1..n} j*a(j)*(Sum_{i=1..n-j} a(i)*a(n-j-i)))/n for n >= 1, where a(k) = 0 if k not an integer (essentially eq (4) in the Robinson et al. paper). - Emeric Deutsch, May 16 2004
a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.346304267394183622435... (see A239810). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

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..

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).

A176460 Decimal expansion of (20+2*sqrt(105))/5.

Original entry on oeis.org

8, 0, 9, 8, 7, 8, 0, 3, 0, 6, 3, 8, 3, 8, 3, 9, 3, 5, 3, 2, 8, 8, 4, 1, 5, 4, 7, 2, 2, 0, 8, 4, 2, 0, 7, 9, 6, 2, 9, 4, 0, 1, 3, 0, 6, 5, 3, 8, 1, 9, 3, 3, 1, 7, 1, 8, 1, 6, 7, 9, 1, 3, 9, 9, 9, 5, 6, 1, 3, 9, 1, 9, 4, 2, 8, 2, 1, 4, 1, 6, 2, 9, 1, 7, 0, 8, 9, 2, 6, 5, 1, 3, 5, 1, 4, 1, 8, 6, 9, 4, 7, 8, 1, 7, 8

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of (20+2*sqrt(105))/5 is A010733.

			(20+2*sqrt(105))/5 = 8.09878030638383935328...

Cf. A176461 (decimal expansion of sqrt(105)), A010733 (repeat 8, 10).

evalf((20+2*sqrt(105))/5,100); # Wesley Ivan Hurt, Feb 11 2017

A176535 Decimal expansion of (10 + sqrt(105))/2.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 7, 5, 3, 8, 2, 9, 7, 9, 7, 9, 9, 1, 9, 1, 6, 1, 0, 5, 1, 9, 3, 4, 0, 2, 6, 0, 5, 2, 5, 9, 9, 5, 3, 6, 7, 5, 1, 6, 3, 3, 1, 7, 2, 7, 4, 1, 6, 4, 6, 4, 7, 7, 0, 9, 8, 9, 2, 4, 9, 9, 4, 5, 1, 7, 3, 9, 9, 2, 8, 5, 2, 6, 7, 7, 0, 3, 6, 4, 6, 3, 6, 1, 5, 8, 1, 4, 1, 8, 9, 2, 7, 3, 3, 6, 8, 4, 7, 7, 2

Offset: 2

Klaus Brockhaus, Apr 24 2010

Continued fraction expansion of (10 + sqrt(105))/2 is A010733 preceded by 10.

			(10 + sqrt(105))/2 = 10.12347538297979919161...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000

Cf. A176461 (decimal expansion of sqrt(105)), A010733 (repeat 8, 10).

Digits:=140: evalf(10+sqrt(105))/2; # Wesley Ivan Hurt, Jan 21 2017

RealDigits[(10+Sqrt[105])/2,10,120][[1]] (* Harvey P. Dale, Mar 13 2011 *)

cp's OEIS Frontend

A000625 Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account.

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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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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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A176460 Decimal expansion of (20+2*sqrt(105))/5.

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A176535 Decimal expansion of (10 + sqrt(105))/2.

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