A000602 -id:A000602 - cp's OEIS frontend

A112619 Number of connected simple graphs with n vertices, n+2 edges, and vertex degrees no more than 4.

0, 0, 0, 1, 4, 18, 79, 326, 1278, 4875, 17978, 64720, 227842, 787546, 2678207, 8982754, 29761361, 97558039, 316778169, 1019996738, 3259673935, 10347077497, 32644696187, 102425388286, 319754805262

Offset: 1

Jonathan Vos Post, Dec 21 2005

nonn

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 3 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. [gives different a(18)]

The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, A112410, this sequence, A112408, A112424, A112425, A112426, A112442. Cf. A121941.

for n in {4..15}; do geng -c -D4 ${n} $((n+2)):$((n+2)) -u; done # Andrey Zabolotskiy, Nov 24 2017

Corrected offset and new name from Andrey Zabolotskiy, Nov 24 2017

a(18) corrected and a(19)-a(25) added by Georg Grasegger, Jun 05 2023

A000600 Number of tertiary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 7, 17, 40, 102, 249, 631, 1594, 4074, 10443, 26981, 69923, 182158, 476141, 1249237, 3287448, 8677074, 22962118, 60915508, 161962845, 431536102, 1152022025, 3081015684, 8253947104, 22147214029, 59514474967

Offset: 0

N. J. A. Sloane

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Handbook of Combinatorics, North-Holland '95, p. 1963.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
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).

H. R. Henze, C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
H. R. Henze, C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3045. (Annotated scanned copy)
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly] (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. 30, Eq. 2.6.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.

Cf. A000598.

Henze and Blair give a recurrence; also g.f. A(x) = x*cycle_index(S3, B(x)-1), B(x) = g.f. for A000598.

A002986 Number of non-cyclic hydrocarbons with n carbon atoms (excluding stereoisomers).

Original entry on oeis.org

1, 3, 4, 12, 27, 84, 247, 826, 2777, 9868, 35579, 131847, 495671, 1893819, 7320954, 28619581, 112923053, 449343946, 1801330288, 7269849395, 29517342098, 120507480668, 494449558111, 2038073860257, 8436185990286, 35055744550563, 146195133355612, 611723431211193

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of hypothetical acyclic hydrocarbons with n carbon atoms that satisfy the octet rule. - Natan Arie Consigli, Dec 26 2016

a(n) is the number of acyclic connected multigraphs with n nodes of degree less than 5, except for a(2). - Natan Arie Consigli, May 25 2017

			a(3) = 4 because there are 4 non-cyclic structures that can be drawn with 3 carbons (propane, propene, propyne, and allene). - _David Consiglio, Jr._, May 15 2014

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

Andrew Howroyd, Table of n, a(n) for n = 1..500
Sean A. Irvine, java program
R. C. Read, Some recent results in chemical enumeration, Preprint, circa 1972. (Annotated scanned copy)
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
Wikipedia, Graph

Cf. A000602 (restriction to alkanes).

\\ here S is MSET_k comb class of g
S(g,n,k)={polcoeff(exp( sum(i=1, k, (y^i + O(y*y^k))*subst(g + O(x*x^(n\i)), x, x^i)/i )), k, y) + O(x*x^n)}
R(n)={my(f,g,h); f=g=h=O(x); for(n=1, n, h = x*(1+f); g = h + x*(S(f,n,2) + g); f = g + x*(S(f,n,3) + f*g + h)); [f,g,h]}
seq(n)={my(t=R(n), f=t[1], g=t[2], h=t[3]); Vec(f + x*(S(f, n, 4) + g*S(f, n, 2) + S(g, n, 2) + f*h) + (subst(f+g+h+O(x*x^(n\2)), x, x^2) - f^2 - g^2 - h^2)/2)} \\ Andrew Howroyd, May 26 2018

geng -c -D4 ${n} $[${n}-1]:$[${n}-1] -q | multig -m3 -D4 -u

a(n) ~ c * d^n / n^(5/2), where d = 4.576467424512811226430711636719246756... and c = 0.84315686601314832608482486521039... - Vaclav Kotesovec, Feb 11 2019

Better definition from Sergio Pimentel, Apr 28 2006
a(11) (computed using Nauty) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 24 2008
a(12)-a(13) (computed using Molgen 3.5) from David Consiglio, Jr., May 15 2014
Existing terms verified and a(14)-a(16) from Sean A. Irvine, Dec 22 2014
a(17)-a(19) from Sean A. Irvine, Dec 28 2014
a(18)-a(19) corrected and a(20)-a(24) (computed using nauty) from Sean A. Irvine, Jan 02 2015
Terms a(25) and beyond from Andrew Howroyd, May 26 2018

A018210 Alkane (or paraffin) numbers l(9,n).

Original entry on oeis.org

1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520, 4032, 6216, 9324, 13608, 19440, 27192, 37389, 50556, 67408, 88660, 115258, 148148, 188552, 237692, 297115, 368368, 453376, 554064, 672792, 811920, 974304, 1162800, 1380825, 1631796

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

From M. F. Hasler, May 02 2009: (Start)
Also, 6th column of A159916, i.e., number of 6-element subsets of {1,...,n+6} whose elements add up to an odd integer.
Third differences are A002412([n/2]). (End)
F(1,6,n) is the number of bracelets with 1 blue, 6 identical red and n identical black beads. If F(1,6,1) = 4 and F(1,6,2) = 16 taken as a base, F(1,6,n) = n(n+1)(n+2)(n+3)(n+4)/120 + F(1,4,n) + F(1,6,n-2). F(1,4,n) is the number of bracelets with 1 blue, 4 identical red and n identical black beads. If F(1,4,1) = 3 and F(1,4,2) = 9 taken as a base; F(1,4,n) = n(n+1)(n+2)/6 + F(1,2,n) + F(1,4,n-2). F(1,2,n) is the number of bracelets with 1 blue, 2 identical red and n identical black beads. If F(1,2,1) = 2 and F(1,2,2) = 4 taken as a base F(1,2,n) = n + 1 + F(1,2,n-2). - Ata Aydin Uslu and Hamdi G. Ozmenekse, Mar 16 2012

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
Winston C. Yang (paper in preparation).

N. J. A. Sloane, Classic Sequences
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,6,n)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,4,n)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,2,n)
Index entries for linear recurrences with constant coefficients, signature (4, -3, -8, 14, 0, -14, 8, 3, -4, 1).

Cf. A005995 (first differences).

a:=n-> (Matrix([[1,0$7,3,12]]). Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [4, -3, -8, 14, 0, -14, 8, 3, -4, 1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..33); # Alois P. Heinz, Jul 31 2008

CoefficientList[(1+3*x^2)/((1-x)^7*(1+x)^3) + O[x]^34, x] (* Jean-François Alcover, Jun 08 2015 *)
LinearRecurrence[{4, -3, -8, 14, 0, -14, 8, 3, -4, 1},{1, 4, 16, 44, 110, 236, 472, 868, 1519, 2520},34] (* Ray Chandler, Sep 23 2015 *)

A018210(n)=(n+2)*(n+4)*(n+6)^2*(n^2+3*n+5)/1440-if(n%2,(n^2+7*n+11)/32) \\ M. F. Hasler, May 02 2009

G.f.: (1+3*x^2)/(1-x)^4/(1-x^2)^3. - N. J. A. Sloane
l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
a(2n) = (n+1)(n+2)(n+3)^2(4n^2+6n+5)/90, a(2n-1) = n(n+1)(n+2)(n+3)(4n^2+6n+5)/90. - M. F. Hasler, May 02 2009
a(n) = (1/(2*6!))*(n+2)*(n+4)*(n+6)*((n+1)*(n+3)*(n+5) + 1*3*5) - (1/2)*(1/2^4)*(n^2+7*n+11)*(1/2)*(1-(-1)^n). - Yosu Yurramendi, Jun 23 2013
a(n) = A060099(n)+3*A060099(n-2). - R. J. Mathar, May 08 2020

A018213 Alkane (or paraffin) numbers l(12,n).

Original entry on oeis.org

1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, 408760, 653752, 1021735, 1562275, 2343770, 3453450, 5008003, 7153575, 10080720, 14024400, 19284460, 26225628, 35304920, 47071640

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Equals (1/2) * ((A000582) + (A000332 interleaved with zeros)) = (1/2) * ((1, 10, 55, 220, 715...) + (1, 0, 5, 0, 15,...)); where A000582 = binomial(n,9) and A000332 = binomial(n,4).

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
Winston C. Yang (paper in preparation).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Classic Sequences
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1).

[(1/(2*Factorial(9)))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)+(1/6)*(1/2^7)*(n+2)*(n+4)*(n+6)*(n+8)*(1/2)*(1+(-1)^n): n in [0..40]]; // Vincenzo Librandi, Oct 16 2013

CoefficientList[Series[(5 x^4 + 10 x^2 + 1)/((x - 1)^10 (x + 1)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 16 2013 *)
LinearRecurrence[{5, -5, -15, 35, 1, -65, 45, 45, -65, 1, 35, -15, -5, 5, -1},{1, 5, 30, 110, 365, 1001, 2520, 5720, 12190, 24310, 46252, 83980, 147070, 248710, 408760},101] (* Ray Chandler, Sep 23 2015 *)

l(c, r) = 1/2 binomial(c+r-3, r) + 1/2 d(c, r), where d(c, r) is binomial((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, binomial((c + r - 4)/2, r/2) if c is even and r is even, binomial((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
G.f.: (5*x^4+10*x^2+1)/((x-1)^10*(x+1)^5). [Colin Barker, Aug 06 2012]
a(n) = (1/(2*9!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9) +(1/6)*(1/2^7)*(n+2)*(n+4)*(n+6)*(n+8)*(1/2)*(1+(-1)^n). [Yosu Yurramendi, Jun 23 2013]

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..

A112442 Number of connected simple graphs with n vertices, n+7 edges, and vertex degrees no more than 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 35, 707, 11477, 146428, 1530906, 13663758, 107554370, 764873164, 5004170844, 30537798974, 175688807383, 960958921848, 5030916734826

Offset: 1

Jonathan Vos Post, Dec 21 2005

J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 8 on page 103.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs [gives different a(11)].

The analogs for n+k edges with k = -1, 0, ..., 6 are: A000602, A036671, A112410, A112619, A112408, A112424, A112425, A112426. Cf. A121941.

for n in {7..13}; do geng -c -D4 ${n} $((n+7)):$((n+7)) -u; done # Andrey Zabolotskiy, Nov 24 2017

New name, offset corrected, a(11) corrected, and a(14) added by Andrey Zabolotskiy, Nov 24 2017

a(15)-a(20) added by Georg Grasegger, Jun 05 2023

A204293 Pascal's triangle interspersed with rows of zeros, and the rows of Pascal's triangle are interspersed with zeros.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 15, 0, 20

Offset: 0

Reinhard Zumkeller, Jan 14 2012

Auxiliary array for computing Losanitsch's triangle A034851;

T(n, k) + T(n, k + 2) = T(n + 2, k + 2) for k < n - 1.

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences: Losanitsch's Triangle
Eric Weisstein's World of Mathematics, Losanitsch's Triangle
Index entries for triangles and arrays related to Pascal's triangle

Cf. A077957 (row sums), A126869 (central terms); A108044, A007318.

a204293 n k = a204293_tabl !! n !! k
a204293_row n = a204293_tabl !! n
a204293_tabl = [1] : [0,0] : f [1] [0,0] where
   f xs ys = xs' : f ys xs' where
     xs' = zipWith (+) ([0,0] ++ xs) (xs ++ [0,0])

t[n_?EvenQ, k_?EvenQ] := Binomial[n/2, k/2]; t[, ] = 0; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Feb 07 2012 *)

T(n, k) = (1 - n mod 2) * (1 - k mod 2) * binomial(floor(n/2),floor(k/2)).

Formula for T(n,k) corrected by Peter Bala, Jul 06 2015

A000599 Number of secondary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.

Original entry on oeis.org

0, 0, 1, 1, 3, 6, 15, 33, 82, 194, 482, 1188, 2988, 7528, 19181, 49060, 126369, 326863, 849650, 2216862, 5806256, 15256265, 40210657, 106273050, 281593237, 747890675, 1990689459, 5309397294, 14187485959, 37977600390, 101827024251

Offset: 1

N. J. A. Sloane

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Handbook of Combinatorics, North-Holland '95, p. 1963.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
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).

H. R. Henze, C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
H. R. Henze, C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3045. (Annotated scanned copy)
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly] (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, 38 (Q(x)).
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.

Cf. A000598, A000600.

Henze and Blair give a recurrence.; also g.f. A(x) = x*cycle_index(S2, B(x)-1), where B(x) is g.f. for A000598.

A000678 Number of carbon (rooted) trees with n carbon atoms = unordered 4-tuples of ternary trees.

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 18, 42, 96, 229, 549, 1347, 3326, 8330, 21000, 53407, 136639, 351757, 909962, 2365146, 6172068, 16166991, 42488077, 112004630, 296080425, 784688263, 2084521232, 5549613097, 14804572332, 39568107511, 105938822149

Offset: 0

N. J. A. Sloane, E. M. Rains (rains(AT)caltech.edu)

			z+z^2+2*z^3+4*z^4+9*z^5+18*z^6+42*z^7+...

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 454).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 527.
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..60
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 10 of Table I.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 10 (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
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 g.f. called P(x) on p. 28, 37.
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Let T_i(z) = g.f. for ternary trees of height at most i.
N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3)+O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: # G000598 = g.f. for A000598
i := 0: while iA000678 := n->coeff(G000678,z,n); # G000678 = g.f. for A000678.
(this Maple program continues in A000022.)

m = 45; (* T = G000598 *) T[] = 0; Do[T[z] = 1 + z*(T[z]^3/6 + T[z^2]*T[z]/2 + T[z^3]/3) + O[z]^m // Normal, m];
G000678[z_] = z*(T[z]^4/24 + T[z^2]*T[z]^2/4 + T[z^2]^2/8 + T[z]*T[z^3]/3 + T[z^4]/4) + O[z]^m;
CoefficientList[G000678[z], z] (* Jean-François Alcover, Jan 11 2018, after N. J. A. Sloane *)

G.f.: A(x) = x*cycle_index(S4, B(x)), B(x) = g.f. for A000598.

cp's OEIS Frontend

A112619 Number of connected simple graphs with n vertices, n+2 edges, and vertex degrees no more than 4.

Views

Author

Keywords

Links

Crossrefs

Programs

nauty

Extensions

A000600 Number of tertiary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.

Views

Author

Keywords

References

Links

Crossrefs

Formula

A002986 Number of non-cyclic hydrocarbons with n carbon atoms (excluding stereoisomers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

nauty

Formula

Extensions

A018210 Alkane (or paraffin) numbers l(9,n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A018213 Alkane (or paraffin) numbers l(12,n).

Views

Author

Keywords

Comments

References

Links

Programs

Magma

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A112442 Number of connected simple graphs with n vertices, n+7 edges, and vertex degrees no more than 4.

Views

Author

Keywords

Links

Crossrefs

Programs

nauty

Extensions

A204293 Pascal's triangle interspersed with rows of zeros, and the rows of Pascal's triangle are interspersed with zeros.

Views

Author

Keywords

Comments