A000623 -id:A000623 - 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

A000621 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are not stereoisomers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 14, 23, 39, 65, 110, 184, 310, 520, 876, 1471, 2475, 4159, 6996, 11759, 19775, 33244, 55902, 93984, 158030, 265696, 446746, 751128, 1262940, 2123444, 3570318, 6002983, 10093259, 16970431, 28533590, 47975381, 80664329, 135626284, 228037752

Offset: 1

N. J. A. Sloane

Also number of monosubstituted alkanes C(n)H(2n+1)-X of the form R-CH2-X (primary) that are not stereoisomers.
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) = this sequence, Ss = A000622, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 14*x^8 + 23*x^9 + ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 300.
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. [This replaces an earlier b-file computed by Vincenzo Librandi (and corrected terms 64-1000).]
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 (1932), 1098-1105.
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. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 283
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, "q" on page 441.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, "q" on page 441. (Annotated scanned copy)
G. Polya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, vol.68, no.1, pp.145-254, (1937). (see pp.151-152).

Cf. A000620-A000625, A239804, A239806.

nmax=40; a=1-x; Do[a=1/(1-x (a/.x->x^2)),{Log[2,nmax]+2}]; CoefficientList[Series[a,{x,0,nmax-1}],x] (* Jean-François Alcover, Jun 16 2011, after Michael Somos, fixed by Vaclav Kotesovec, Mar 28 2014 *)
max = 40; cf = Fold[Function[1 - x^#2/#1], 1, 2^Reverse[Range[0, Floor[Log[2, max]]]]]; List @@ (1-Series[cf, {x, 0, 2*max}] // Normal) /. x -> 1 (* Jean-François Alcover, Sep 24 2014 *)

T(n,m):=if m=n then 1 else sum(binomial(i+m-1,i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i),i,1,n-m);
makelist(T(2*n-1,1),n,1,30); /* Vladimir Kruchinin, Mar 18 2015 */

{a(n) = my(A, m); if( n<1, 0, n--; m = 1; A = 1 + O(x); while( m<=n, m *= 2; A = 1 / (1 - x * subst(A, x, x^2)) ); polcoeff( A, n )) }; /* Michael Somos, Sep 03 2007 */

G.f.: A(x) satisfies A(x) = 1/(1-x*A(x^2)), with offset 0. - Paul D. Hanna, Aug 16 2002
Given g.f. A(x), then B(x) = A(x) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (1 - u)^2 * w - u^2 * v * (v - 1). - Michael Somos, Sep 03 2007
G.f.: x / (1 - x / (1 - x^2 / (1 - x^4 / (1 - ...)))). - Michael Somos, Sep 03 2007
From Joerg Arndt, Oct 15 2011: (Start)
For offset 0 (as considered in the 1937 Polya reference) we have
G.f.: A(x) = 1 / (1 - x / (1 - x^2 / (1 - x^4 / (1 - ...)))) and
A(x) satisfies A(x) = 1 + x*A(x)*A(x^2) (equivalent to Hanna's functional equation).
(End)
a(n) ~ c * beta^n, where beta = 1.681367524441880255591... (see A239804), c = 0.214536139134648555630... (see A239806). Asymptotic formula a(n) ~ K * beta^n from reference (Analytic Combinatorics, p. 283), where K = 0.3607140971, beta = 1.6813675244^n is for offset 0 (beta is same, but K = c * beta = 0.360714097160142828...). - Vaclav Kotesovec, Mar 27 2014
a(n) = T(2*n-1,1), where T(n,m) = Sum_{i=1..n-m} binomial(i+m-1,i)*((1+(-1)^(n-m))/2)*T((n-m)/2,i), n > m, T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015
a(n) = A253190(2*n-1,1). - R. J. Mathar, Dec 16 2015

Additional comments from Bruce Corrigan, Nov 04 2002

Formulae edited by N. J. A. Sloane, Feb 27 2006

A000620 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are stereoisomers.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 20, 60, 176, 512, 1488, 4326, 12648, 37186, 109980, 327216, 979020, 2944414, 8897732, 27005290, 82288516, 251650788, 772127678, 2376238138, 7333188770, 22688297950, 70360977228, 218678818026, 681017928476, 2124840874610, 6641336507270, 20791999731518

Offset: 1

N. J. A. Sloane

nonn

Also number of monosubstituted alkanes C(n)H(2n+1)-X of the form R-CH2-X (primary) that are stereoisomers.
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) = this sequence, Pn (and An, Sn) = A000621, Ss = A000622, 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
Jean-François Alcover, Mathematica program
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. A000621, A000622, A000623, A000624, A000625, A239803, A239805.

# Blair and Henze's recurrences for A000620-A000625 (see comments lines for relationship between the sequences and their symbols).
Ps := [0,0,0]; Pn := [1,1,1]; Ss := [0,0,0]; Sn := [0,0,1]; Ts := [0,0,0]; Tn := [0,0,0]; As := [0,0,0]; An := [1,1,2]; T := [1,1,2];
for n from 4 to 60 do Ps := [op(Ps),As[n-1]]; Pn := [op(Pn),An[n-1]]; t1 := add( 2*T[n-1-j]*T[j],j=1..floor((n-2)/2) ); if n mod 2 = 1 then i := (n-1)/2; t1 := t1+T[i]^2-An[i]; fi; Ss := [op(Ss),t1];
t2 := 0; if n mod 2 = 1 then i := (n-1)/2; t2 := An[i]; fi; Sn := [op(Sn),t2]; t3 := 0; for i from 1 to (n-1)/3 do for j from i+1 to (n-2)/2 do k := n-1-i-j; if j 0 and i <> j then t4 := t4+(T[i]^2-An[i])*T[j]+An[i]*As[j]; t5 := t5+An[i]*An[j]; fi; od; t6 := 0; t7 := 0; if n mod 3 = 1 then i := (n-1)/3; t6 := (2*T[i]+T[i]^3)/3-An[i]^2; t7 := An[i]^2; fi;
Ts := [op(Ts), t3+t4+t6]; Tn := [op(Tn), t5+t7]; As := [op(As), Ps[n]+Ss[n]+Ts[n]]; An := [op(An), Pn[n]+Sn[n]+Tn[n]]; T := [op(T),As[n]+An[n]]; od: Ps; Pn; Ss; Ts; Tn; T;

Mathematica
```
(* See links *)
```

See Maple program for recurrences for this sequence and A000621-A000625.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.105352133282419523497... (see A239805). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A000622 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, 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

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

cp's OEIS Frontend

A239803 Decimal expansion of a constant related to A000620, A000622, A000623 and A000625.

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A239808 Decimal expansion of a constant related to A000623.

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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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A000621 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are not stereoisomers.

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A000620 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are stereoisomers.

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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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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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