A036370 -id:A036370 - cp's OEIS frontend

A036437 Triangle of coefficients of generating function of ternary rooted trees of height exactly n.

1, 1, 1, 1, 1, 2, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 3, 8, 15, 27, 43, 67, 97, 136, 183, 239, 300, 369, 432, 498, 551, 594, 614, 624, 601, 570, 514, 453, 378, 312, 238, 181, 128, 89, 56, 37, 20, 12, 6, 3, 1, 1, 1, 4, 13, 32, 74, 155, 316, 612, 1160, 2126, 3829, 6737

Offset: 1

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

			1;
1, 1, 1;
1, 2, 4, 4, 5, 4, 4, 3, 2, 1, 1;

Alois P. Heinz, Rows n = 1..8, flattened
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
Index entries for sequences related to rooted trees

df:= (t, l)-> zip((x,y)->x-y, t, l, 0):
T:= proc(n) option remember; local f, g;
      if n=0 then 1
    else f:= z-> add([T(n-1)][i]*z^(i-1), i=1..nops([T(n-1)]));
         g:= expand(1 +z*(f(z)^3/6 +f(z^2)*f(z)/2 +f(z^3)/3));
         seq(coeff(g, z, i), i=0..degree(g, z))
      fi
    end:
seq(df([T(n)], [T(n-1)])[n+1..-1][], n=1..5); # Alois P. Heinz, Sep 26 2011

df[t_, l_] := Plus @@ PadRight[{t, -l}]; T[n_] := T[n] = Module[{f, g}, If[n == 0, {1}, f[z_] := Sum[T[n-1][[i]]*z^(i-1), {i, 1, Length[T[n-1]]}]; g = Expand[1+z*(f[z]^3/6+f[z^2]*f[z]/2+f[z^3]/3)]; Table [Coefficient [g, z, i], {i, 0, Exponent[g, z]}]]]; Table[df[T[n], T[n-1]][[n+1 ;; -1]], {n, 1, 5}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

T_{n}(z) - T_{n-1}(z) (see A036370).

A036373 Number of ternary rooted trees with n nodes and height at most 5.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 33, 63, 121, 225, 415, 749, 1344, 2365, 4129, 7106, 12104, 20354, 33883, 55706, 90628, 145729, 231801, 364555, 567206, 872727, 1328545, 2000536, 2980554, 4393287, 6407683, 9246830, 13204526, 18657905, 26088244

Offset: 0

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

Sean A. Irvine, Table of n, a(n) for n = 0..121
Index entries for sequences related to rooted trees

Cf. A036370.

T[0] = {1}; T[n_] := T[n] = Module[{f, g}, f[z_] := Sum[T[n - 1][[i]]*z^(i - 1), {i, 1, Length[T[n - 1]]}]; g = 1 + z*(f[z]^3/6 + f[z^2]*f[z]/2 + f[z^3]/3); CoefficientList[g, z]]; A036373 = T[5] (* Jean-François Alcover, Jan 19 2016, after Alois P. Heinz (A036370) *)

If T_i(z) = g.f. for ternary trees of height at most i, T_{i+1}(z)=1+z*(T_i(z)^3/6+T_i(z^2)*T_i(z)/2+T_i(z^3)/3); T_0(z) = 1.

A036374 Number of ternary rooted trees with n nodes and height at most 6.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 38, 82, 177, 376, 789, 1638, 3376, 6894, 13987, 28181, 56424, 112282, 222171, 437098, 855311, 1664755, 3223402, 6209505, 11901967, 22700056, 43083657, 81376732, 152971812, 286199220, 532954482, 987861697, 1822655134

Offset: 0

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

Sean A. Irvine, Table of n, a(n) for n = 0..364
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.
Index entries for sequences related to rooted trees

Cf. A036370.

T[0] = {1}; T[n_] := T[n] = Module[{f, g}, f[z_] := Sum[T[n - 1][[i]]*z^(i - 1), {i, 1, Length[T[n - 1]]}]; g = 1 + z*(f[z]^3/6 + f[z^2]*f[z]/2 + f[z^3]/3); CoefficientList[g, z]]; A036374 = T[6] (* Jean-François Alcover, Jan 19 2016, after Alois P. Heinz (A036370) *)

If T_i(z) = g.f. for ternary trees of height at most i, T_{i+1}(z)=1+z*(T_i(z)^3/6+T_i(z^2)*T_i(z)/2+T_i(z^3)/3); T_0(z) = 1.

A036375 Number of ternary rooted trees with n nodes and height at most 7.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 39, 88, 203, 464, 1056, 2381, 5344, 11900, 26381, 58165, 127713, 279209, 608213, 1319985, 2855275, 6155981, 13231553, 28353787, 60583959, 129084369, 274283708, 581244959, 1228514486, 2589902750, 5446168197

Offset: 0

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

Sean A. Irvine, Table of n, a(n) for n = 0..1093
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.
Index entries for sequences related to rooted trees

Cf. A036370.

T[0] = {1}; T[n_] := T[n] = Module[{f, g}, f[z_] := Sum[T[n - 1][[i]]*z^(i - 1), {i, 1, Length[T[n - 1]]}]; g = 1 + z*(f[z]^3/6 + f[z^2]*f[z]/2 + f[z^3]/3); CoefficientList[g, z]]; A036375 = T[7] (* Jean-François Alcover, Jan 19 2016, after Alois P. Heinz (A036370) *)

If T_i(z) = g.f. for ternary trees of height at most i, T_{i+1}(z)=1+z*(T_i(z)^3/6+T_i(z^2)*T_i(z)/2+T_i(z^3)/3); T_0(z) = 1.

A036372 Number of ternary rooted trees with n nodes and height at most 4.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 12, 20, 31, 47, 70, 99, 137, 184, 239, 300, 369, 432, 498, 551, 594, 614, 624, 601, 570, 514, 453, 378, 312, 238, 181, 128, 89, 56, 37, 20, 12, 6, 3, 1, 1

Offset: 0

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

Index entries for sequences related to rooted trees

Cf. A036370.

T[0] = {1}; T[n_] := T[n] = Module[{f, g}, f[z_] := Sum[T[n - 1][[i]]*z^(i - 1), {i, 1, Length[T[n - 1]]}]; g = 1 + z*(f[z]^3/6 + f[z^2]*f[z]/2 + f[z^3]/3); CoefficientList[g, z]]; A036372 = T[4] (*Jean-François Alcover, Jan 19 2016, after Alois P. Heinz (A036370) *)

If T_i(z) = g.f. for ternary trees of height at most i, T_{i+1}(z)=1+z*(T_i(z)^3/6+T_i(z^2)*T_i(z)/2+T_i(z^3)/3); T_0(z) = 1.

A036376 Number of ternary rooted trees with n nodes and height at most 8.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 39, 89, 210, 498, 1185, 2813, 6664, 15707, 36886, 86259, 201003, 466788, 1080825, 2495565, 5747664, 13206253, 30276788, 69267205, 158155198, 360422113, 819873747, 1861732042, 4220347570, 9551287776

Offset: 0

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

Sean A. Irvine, Table of n, a(n) for n = 0..3280
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.
Index entries for sequences related to rooted trees

Cf. A036370.

T[0] = {1}; T[n_] := T[n] = Module[{f, g}, f[z_] := Sum[T[n - 1][[i]]*z^(i - 1), {i, 1, Length[T[n - 1]]}]; g = 1 + z*(f[z]^3/6 + f[z^2]*f[z]/2 + f[z^3]/3); CoefficientList[g, z]]; A036376 = T[8][[1 ;; 40]] (* Jean-François Alcover, Jan 19 2016, after Alois P. Heinz (A036370) *)

If T_i(z) = g.f. for ternary trees of height at most i, T_{i+1}(z)=1+z*(T_i(z)^3/6+T_i(z^2)*T_i(z)/2+T_i(z^3)/3); T_0(z) = 1.

A287211 The number of plane rooted complete ternary trees with 2n+1 unlabeled leaves (hence n internal nodes including the root where n starts at 0) satisfying these two conditions: (1) if one of the three children of any internal node is the greatest in deglex order then that child is not the leftmost child; (2) if one of the three children of any internal node is the smallest in deglex order then that child is not the rightmost child. Deglex order refers to degree-lexicographical order defined inductively on the number of leaves (see details under Comments).

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 308, 1264, 5332, 22994, 100896, 449004

Offset: 0

Murray R. Bremner, May 21 2017

"Plane" means "embedded in the plane" or (equivalently) the three children of each internal node (including the root) are ordered left, middle, right. Deglex order on trees with 2n+1 leaves is defined as follows: to compare two such trees T and U with children T_1, T_2, T_3 and U_1, U_2, U_3, first find the least index 1 <= i <= 3 for T_i <> U_i, then compare T_i and U_i in deglex order already defined inductively on trees with fewer than 2n+1 leaves; note that this requires comparing trees with different numbers of leaves, so we say that T_i precedes U_i if either (i) T_i has fewer leaves than U_i, or (ii) T_i and U_i have the same number of leaves, and T_i precedes U_i in deglex order.
An alternative description of this sequence: it counts the distinct association types in arity 2n+1 for a ternary operation [a,b,c] satisfying the cyclic-sum relation [a,b,c] + [b,c,a] + [c,a,b] = 0. The two conditions stated under "Name" are necessary to deal with the possibility of repeated factors: [a,a,b], [a,b,a], [b,a,a] where a < b in deglex order, and [a,b,b], [b,a,b], [b,b,a] where a < b in deglex order.
See further details in the comments to the Maple program which is attached as a a-file.

			Association types for arities 1, 3, 5, 7 are as follows in deglex order. See Links for a-file with association types for arities up to 11.
Arity 1, number of types 1:
a.
Arity 3, number of types 1:
[abc].
Arity 5, number of types 2:
[ab[cde]],
[a[bcd]e].
Arity 7, number of types 6:
[ab[cd[efg]]],
[ab[c[def]g]],
[a[bcd][efg]],
[a[bc[def]]g],
[a[b[cde]f]g],
[[abc]d[efg]].

Murray R. Bremner, Association types up to and including arity 11
Murray R. Bremner, Maple worksheet for generating association types

Cf. A357538, A000598, A000625, A001764, A006629, A036370, A036437, A233389.

Maple
```
See attached a-file under Links.
```

A036371 Number of ternary rooted trees with n nodes and height at most 3.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1

Offset: 0

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

Index entries for sequences related to rooted trees

Cf. A036370.

T[0] = {1}; T[n_] := T[n] = Module[{f, g}, f[z_] := Sum[T[n - 1][[i]]*z^(i - 1), {i, 1, Length[T[n - 1]]}]; g = 1 + z*(f[z]^3/6 + f[z^2]*f[z]/2 + f[z^3]/3); CoefficientList[g, z]]; A036371 = T[3] (* Jean-François Alcover, Jan 19 2016, after Alois P. Heinz (A036370) *)

If T_i(z) = g.f. for ternary trees of height at most i, T_{i+1}(z)=1+z*(T_i(z)^3/6+T_i(z^2)*T_i(z)/2+T_i(z^3)/3); T_0(z) = 1.

A036420 Number of ternary rooted trees with n nodes and height exactly 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 13, 32, 74, 155, 316, 612, 1160, 2126, 3829, 6737, 11672, 19856, 33332, 55112, 90014, 145105, 231200, 363985, 566692, 872274, 1328167, 2000224, 2980316, 4393106, 6407555, 9246741, 13204470, 18657868, 26088224, 36095090

Offset: 0

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

Sean A. Irvine, Table of n, a(n) for n = 0..121
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
Index entries for sequences related to rooted trees

Cf. A036370, A036437.

a(n) = A036373(n) - A036372(n). - Sean A. Irvine, Oct 31 2020

A036421 Number of ternary rooted trees with n nodes and height exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 5, 19, 56, 151, 374, 889, 2032, 4529, 9858, 21075, 44320, 91928, 188288, 381392, 764683, 1519026, 2991601, 5844950, 11334761, 21827329, 41755112, 79376196, 149991258, 281805933, 526546799, 978614867, 1809450608

Offset: 0

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

Sean A. Irvine, Table of n, a(n) for n = 0..364
Index entries for sequences related to rooted trees

Cf. A036370, A036437.

a(n) = A036374(n) - A036373(n). - Sean A. Irvine, Oct 31 2020

cp's OEIS Frontend

A036437 Triangle of coefficients of generating function of ternary rooted trees of height exactly n.

Views

Author

Keywords

Examples

Links

Programs

Maple

Mathematica

Formula

A036373 Number of ternary rooted trees with n nodes and height at most 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A036374 Number of ternary rooted trees with n nodes and height at most 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A036375 Number of ternary rooted trees with n nodes and height at most 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A036372 Number of ternary rooted trees with n nodes and height at most 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A036376 Number of ternary rooted trees with n nodes and height at most 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A036371 Number of ternary rooted trees with n nodes and height at most 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A036420 Number of ternary rooted trees with n nodes and height exactly 5.

Views

Author

Keywords

Links

Crossrefs