A000602 -id:A000602 - cp's OEIS frontend

A018211 Alkane (or paraffin) numbers l(10,n).

1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912, 25236, 38760, 58200, 85272, 122661, 173052, 240460, 328900, 444158, 592020, 780572, 1017900, 1315015, 1682928, 2136304, 2689808, 3362600, 4173840, 5148144, 6310128

Offset: 0

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

nonn

Equals (1/2) * ((1, 8, 36, 120, 330, 792,...) + (1, 0, 4, 0, 10, 0, 20,...)); where (1, 8, 36,..) = A000580 = C(n,7), and (1, 4, 10,...) = the Tetrahedral numbers.

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)
Index entries for linear recurrences with constant coefficients, signature (4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1).

Cf. A282011.

a:= n-> (Matrix([[1, 0$7, -1, -4, -20, -60]]). Matrix(12, (i,j)-> `if`(i=j-1, 1, `if`(j=1, [4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1][i], 0)))^n)[1,1]: seq(a(n), n=0..31); # Alois P. Heinz, Jul 31 2008

LinearRecurrence[{4, -2, -12, 17, 8, -28, 8, 17, -12, -2, 4, -1},{1, 4, 20, 60, 170, 396, 868, 1716, 3235, 5720, 9752, 15912},32] (* Ray Chandler, Sep 23 2015 *)

G.f.: (1+6*x^2+x^4)/((1-x)^4*(1-x^2)^4). [ N. J. A. Sloane ]
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.
a(n) = (1/(2*7!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7) + (1/3)*(1/2^5)*(n+2)*(n+4)*(n+6)*(1/2)*(1+(-1)^n) [Yosu Yurramendi Jun 23 2013]

A034877 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 2, 3, 6, 10, 6, 3, 3, 9, 16, 16, 9, 3, 4, 12, 28, 32, 28, 12, 4, 4, 16, 40, 60, 60, 40, 16, 4, 5, 20, 60, 100, 126, 100, 60, 20, 5, 5, 25, 80, 160, 226, 226, 160, 80, 25, 5, 6, 30, 110, 240, 396, 452, 396, 240, 110, 30, 6, 6, 36, 140, 350, 636, 848

Offset: 0

N. J. A. Sloane

Same as A034852, but omitting the border of 0's.

			Triangle begins:
  1;
  1, 1;
  2, 2,  2;
  2, 4,  4,  2;
  3, 6, 10,  6, 3;
  3, 9, 16, 16, 9, 3;
  ...

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

Reinhard Zumkeller, Rows n=0..25 of triangle, flattened
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
Johann Cigler, Some Pascal-like triangles, 2018.
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
Wikipedia, Losanitsch's triangle

Cf. A007318, A034851, A034852.

Row sums are essentially A032085. Central column is A032095.

a034877 n k = a034877_tabl !! n !! k
a034877_row n = a034877_tabl !! n
a034877_tabl = map (init . tail) $ drop 2 a034852_tabl
-- Reinhard Zumkeller, Dec 16 2013

More terms from James Sellers, May 04 2000

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).

A125064 Number of simple graphs on at most 16 unlabeled vertices with maximal degree at most 4 with a single cycle of length 16-n.

Original entry on oeis.org

1, 2, 11, 39, 169, 534, 1612, 3894, 8771, 16307, 29391, 43291, 69429, 83571

Offset: 0

Parthasarathy Nambi, Jan 05 2007

In the terms of the paper by Hendrickson and Parks, a(n) is the number of monocyclic skeletons with up to 16 nodes with a ring of size 16-n.

James B. Hendrickson and Camden A. Parks, Generation and Enumeration of Carbon Skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), pp. 101-107. See Table VI on page 103.

Cf. A000602, A036671, A112410, A112619, A112408, A112424, A112425, A112426.

Sum_n a(n) = Sum_{k=3..16} A036671(k).

Edited by Andrey Zabolotskiy, Feb 02 2025

A006010 Number of paraffins (see Losanitsch reference for precise definition).

Original entry on oeis.org

1, 5, 20, 52, 117, 225, 400, 656, 1025, 1525, 2196, 3060, 4165, 5537, 7232, 9280, 11745, 14661, 18100, 22100, 26741, 32065, 38160, 45072, 52897, 61685, 71540, 82516, 94725, 108225, 123136, 139520, 157505, 177157, 198612, 221940, 247285, 274721, 304400

Offset: 1

N. J. A. Sloane

This is also the square of the sum of the odd numbers plus the square of the sum of the even numbers, up to n. E.g., a(4) = (1+3)^2 + (2+4)^2 = 52. - Scott R. Shannon, Feb 20 2019

The area of a square whose side is a segment connecting the ends of a broken line (snake), the adjacent links of which are perpendicular and equal to the numbers 1, 2, 3, 4, ..., n. For example, a(5) = 9^2 + 6^2 = 117. - Nicolay Avilov, Aug 02 2022

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nicolay Avilov, Illustration of a(1)-a(6)
Nicolay Avilov, The problem of a broken line in a square (in Russian).
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
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 (3,-1,-5,5,1,-3,1).

Cf. A005994, A186424 (2nd differences), A317614 (1st differences), A335648 (partial sums).

CoefficientList[Series[-(x^4 + 2 x^3 + 6 x^2 + 2 x + 1)/((x - 1)^5 (x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,5,20,52,117,225,400},40] (* Harvey P. Dale, Dec 13 2018 *)

Vec(-x*(x^4+2*x^3+6*x^2+2*x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 05 2015

Sum of [ 1, 3, 9, ... ](A005994) + [ 0, 0, 1, 3, 9, ... ] + 2*[ 0, 1, 5, 15, 35, ... ](binomial(n, 4)).
If n is odd then a(n) = (1/8) * (n^4 + 2*n^3 + 2*n^2 + 2*n + 1) = Det(Transpose[M]*M) where M is the 2 X 3 matrix whose rows are [(n-1)/2, (n-1)/2], [(n-1)/2 + 1, 0] and [(n-1)/2 + 1, (n-1)/2 + 1]. If n is even then a(n) = (1/8) * (n^4 + 2*n^3 + 2*n^2) = Det(Transpose[M]*M) where M is the 2 X 3 matrix whose rows are [n/2, 0], [n/2, n/2] and [n/2 + 1, 0]. - Gerald McGarvey, Oct 30 2007
G.f.: -x*(x^4+2*x^3+6*x^2+2*x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Mar 20 2013
E.g.f.: (x*(7 + 15*x + 8*x^2 + x^3)*cosh(x) + (1 + 5*x + 15*x^2 + 8*x^3 + x^4)*sinh(x))/8. - Stefano Spezia, Jul 08 2020

More terms from David W. Wilson

A018214 Alkane (or paraffin) numbers l(13,n).

Original entry on oeis.org

1, 6, 36, 146, 511, 1512, 4032, 9752, 21942, 46252, 92504, 176484, 323554, 572264, 981024, 1634776, 2656511, 4218786, 6562556, 10016006, 15024009, 22177584, 32258304, 46282704, 65567164, 91792792, 127097712, 174169352

Offset: 0

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

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 (6, -10, -10, 50, -34, -66, 110, 0, -110, 66, 34, -50, 10, 10, -6, 1).

[(1/(2*Factorial(10)))*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*((n+1)*(n+3)*(n+5)*(n+7)*(n+9)+1*3*5*7*9)-(1/6)*(1/2^8)*(n^4+22*n^3+170*n^2+539*n+579)*(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)^11 (x + 1)^5), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 16 2013 *)
LinearRecurrence[{6, -10, -10, 50, -34, -66, 110, 0, -110, 66, 34, -50, 10, 10, -6, 1},{1, 6, 36, 146, 511, 1512, 4032, 9752, 21942, 46252, 92504, 176484, 323554, 572264, 981024, 1634776},28] (* Ray Chandler, Sep 23 2015 *)

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.
G.f.: -(5*x^4+10*x^2+1)/((x-1)^11*(x+1)^5). [Colin Barker, Aug 06 2012]
a(n) = (1/(2*10!))*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*((n+1)*(n+3)*(n+5)*(n+7)*(n+9) + 1*3*5*7*9)- (1/6)*(1/2^8)*(n^4+22*n^3+170*n^2+539*n+579)*(1/2)*(1-(-1)^n). [Yosu Yurramendi, Jun 23 2013]

A000673 Number of bicentered 3-valent (or boron, or binary) trees with n nodes.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 2, 6, 8, 18, 30, 67, 127, 275, 551, 1192, 2507, 5475, 11820, 26007, 57077, 126686, 281625, 630660, 1416116, 3195784, 7232624, 16430563, 37429146, 85528079, 195940960, 450074270, 1036226173, 2391193488, 5529420585

Offset: 0

N. J. A. Sloane

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. 451).
R. C. Read, personal communication.
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).

Nicolas Broutin and Philippe Flajolet, The distribution of height and diameter in random non-plane binary trees, Random Struct. Algorithms 41, No. 2, 215-252 (2012).
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, Letter to N. J. A. Sloane, Oct. 29, 1976
Index entries for sequences related to trees

A000672 = A000673 + A000675. Cf. A000022, A000200, A000602.

n = 50; (* algorithm from Rains and Sloane *)
S2[f_,h_,x_] := f[h,x]^2/2 + f[h,x^2]/2;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S2[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* Robert A. Russell, Sep 15 2018 *)

A000675 Number of centered 3-valent (or boron, or binary) trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 4, 5, 10, 19, 36, 68, 138, 277, 581, 1218, 2591, 5545, 12026, 26226, 57719, 127685, 284109, 634919, 1425516, 3212890, 7269605, 16504439, 37592604, 85876345, 196717882, 451768247, 1039990913, 2399476030, 5547849750

Offset: 0

N. J. A. Sloane

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. 451).
R. C. Read, personal communication.
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).

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, Letter to N. J. A. Sloane, Oct. 29, 1976
Index entries for sequences related to trees

A000672 = A000673 + A000675. Cf. A000022, A000200, A000602.

n = 50; (* algorithm from Rains and Sloane *)
S2[f_,h_,x_] := f[h,x]^2/2 + f[h,x^2]/2;
S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S2[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + S3[T,h-1,z]z - S3[T,h-2,z]z - (T[h-1,z] - T[h-2,z]) (T[h-1,z]-1),z], n+1], {h,1,n/2}] + PadRight[{1,1}, n+1] (* Robert A. Russell, Sep 15 2018 *)

A005996 G.f.: 2(1-x^3)/((1-x)^5(1+x)^2).

Original entry on oeis.org

2, 6, 16, 30, 54, 84, 128, 180, 250, 330, 432, 546, 686, 840, 1024, 1224, 1458, 1710, 2000, 2310, 2662, 3036, 3456, 3900, 4394, 4914, 5488, 6090, 6750, 7440, 8192, 8976, 9826, 10710, 11664, 12654, 13718, 14820, 16000, 17220, 18522, 19866, 21296, 22770, 24334

Offset: 1

N. J. A. Sloane

a(n) is also the number of triples (w,x,y) having all terms in {0,...,n} and wClark Kimberling, Jun 10 2012
a(n) is also the sum of all elements of the square matrix M(n-1) = M1(n-1) x M2(n-1), where M1(n) is the square matrix with elements m1(i,j)= (1+(-1)^(i+j+1))/2, A057212; and M2(n) is the square matrix given by m2(i,j)= (1+(-1)^(i+j))/2, A057212. - Enrique Pérez Herrero, Jun 15 2013
Also the number of longest paths in the (n+1)-web graph for n > 2. - Eric W. Weisstein, Mar 27 2018
a(n) also is the number of undirected rook moves on an n X n chessboard, taken up to 180 degree rotation and axial reflections (horizontal and vertical), for n >= 2. - Hilko Koning, Aug 16 2025

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

Enrique Pérez Herrero, Table of n, a(n) for n = 1..1000
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Longest Path Problem
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Essentially twice A034828.

Table[(1/4)*(1 + n)*(-2 + 5*n + n^2 + 2*Ceiling[1/2 - n/2] - 4*Floor[n/2]), {n, 1, 200}] (* Enrique Pérez Herrero, Aug 03 2012 *)
CoefficientList[Series[2 (1 - x^3)/((1 - x)^5 (1 + x)^2), {x, 0, 40}], x] (* Harvey P. Dale, Apr 08 2013 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {2, 6, 16, 30, 54, 84}, 40] (* Harvey P. Dale, Apr 08 2013 *)
Table[(n + 1) (2 n (n + 2) + 1 - (-1)^n)/8, {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *)

a(n) = 2*(A006918(n) + A006918(n-1) + A006918(n-2)), n>1. - Ralf Stephan, Apr 26 2003
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6), with a(1)=2, a(2)=6, a(3)=16, a(4)=30, a(5)=54, a(6)=84. - Harvey P. Dale, Apr 08 2013
From Ayoub Saber Rguez, Nov 20 2021: (Start)
a(n) = A143785(n) - A002620(n+1).
a(n) = A128624(n) + A002620(n+1).
a(n) = (n^3 + 3*n^2 + 2*n + 1 + n*(n mod 2) - ((n+1) mod 2))/4. (End)

Edited by N. J. A. Sloane, Aug 03 2012

A006006 Weight distribution of [ 128,29,32 ] 2nd-order Reed-Muller code.

Original entry on oeis.org

1, 0, 0, 0, 10668, 0, 5291328, 112881664, 300503590, 112881664, 5291328, 0, 10668, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane

			x^128+10668*x^96*y^32+5291328*x^80*y^48+112881664*x^72*y^56+300503590*x^64*y^64+112881664*x^56*y^72+5291328*x^48*y^80+10668*x^32*y^96+y^128
The weight distribution is:
i A_i
0 1
32 10668
48 5291328
56 112881664
64 300503590
72 112881664
80 5291328
96 10668
128 1

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978.

E. R. Berlekamp and N. J. A. Sloane, Weight Enumerator for Second-Order Reed-Muller Codes, IEEE Trans. Information Theory, IT-16 (1970), 745-751.
Claude Carlet and Patrick Solé, The weight spectrum of several families of Reed-Muller codes, arXiv:2301.13497 [cs.IT], 2023.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
M. Terada, J. Asatani and T. Koumoto, Weight Distribution

R := ReedMullerCode(2,7); W := WeightEnumerator(R); W;

cp's OEIS Frontend

A018211 Alkane (or paraffin) numbers l(10,n).

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Maple

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A034877 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

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Haskell

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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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A125064 Number of simple graphs on at most 16 unlabeled vertices with maximal degree at most 4 with a single cycle of length 16-n.

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A006010 Number of paraffins (see Losanitsch reference for precise definition).

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Mathematica

PARI

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A018214 Alkane (or paraffin) numbers l(13,n).

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Magma

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A000673 Number of bicentered 3-valent (or boron, or binary) trees with n nodes.

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Mathematica

A000675 Number of centered 3-valent (or boron, or binary) trees with n nodes.

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A005996 G.f.: 2(1-x^3)/((1-x)^5(1+x)^2).