cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A120981 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 1 (n >= 0, k >= 0).

Original entry on oeis.org

1, 0, 3, 3, 0, 9, 1, 27, 0, 27, 18, 12, 162, 0, 81, 15, 270, 90, 810, 0, 243, 138, 270, 2430, 540, 3645, 0, 729, 189, 2898, 2835, 17010, 2835, 15309, 0, 2187, 1218, 4536, 34776, 22680, 102060, 13608, 61236, 0, 6561, 2280, 32886, 61236, 312984, 153090, 551124
Offset: 0

Views

Author

Emeric Deutsch, Jul 21 2006

Keywords

Comments

A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

Examples

			T(2,0)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
   1;
   0,   3;
   3,   0,   9;
   1,  27,   0,  27;
  18,  12, 162,   0, 81;
  15, 270,  90, 810,  0, 243;

Links

Crossrefs

Diagonals include A129530, A036216.
Cf. A001764 (row sums), A004319, A120429, A120982, A120983, A120984.

Programs

  • Maple
    T:=proc(n,k) if k<=n then (1/(n+1))*binomial(n+1,k)*sum(3^(3*j-n+2*k)*binomial(n+1-k,j)*binomial(j,n-k-2*j),j=0..n+1-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
  • Mathematica
    T[n_, k_] := (1/(n+1))*Binomial[n+1, k]*Sum[3^(2k - n + 3j)*Binomial[n + 1 - k, j]*Binomial[j, n - k - 2j], {j, 0, n - k + 1}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2018 *)
  • PARI
    T(n,k) = binomial(n+1, k)*sum(j=0, n+1-k, 3^(2*k-n+3*j)*binomial(n+1-k, j)*binomial(j, n-k-2*j))/(n+1); \\ Andrew Howroyd, Nov 06 2017
  • Python
    from sympy import binomial
def T(n, k): return binomial(n + 1, k)*sum([3**(2*k - n + 3*j)*binomial(n + 1 - k, j)*binomial(j, n - k - 2*j) for j in range(n + 2 - k)])//(n + 1)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Nov 07 2017

Formula

T(n,0) = A120984(n).
Sum_{k>=1} k*T(n,k) = 3*binomial(3n,n-1) = 3*A004319(n).
T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k} 3^(2k-n+3j)*binomial(n+1-k,j)*binomial(j,n-k-2j).
G.f.: G=G(t,z) satisfies G = 1 + 3tzG + 3z^2*G^2 + z^3*G^3.

A120982 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 2 (n >= 0, k >= 0).

Original entry on oeis.org

1, 3, 9, 3, 28, 27, 93, 162, 18, 333, 825, 270, 1272, 3915, 2430, 135, 5085, 18144, 17199, 2835, 20925, 84000, 106596, 34020, 1134, 87735, 391554, 612360, 308448, 30618, 372879, 1838295, 3369600, 2364390, 459270, 10206, 1602450, 8674380
Offset: 0

Views

Author

Emeric Deutsch, Jul 21 2006

Keywords

Comments

A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

Examples

			T(2,1)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
    1;
    3;
    9,   3;
   28,  27;
   93, 162,  18;
  333, 825, 270;

Crossrefs

Cf. A001764, A003408, A120429, A120981, A120983, A120985.

Programs

  • Maple
    T:=(n,k)->(1/(n+1))*binomial(n+1,k)*sum(3^(n-k-3*q)*binomial(n+1-k,k+1+2*q)*binomial(n-2*k-2*q,q),q=0..n/2-k):for n from 0 to 12 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

Formula

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..floor(n/2)-k} 3^(n-k-3j)*binomial(n+1-k, k+1+2j)*binomial(n-2k-2j, j).
G.f.: G = G(t,z) satisfies G = 1 + 3zG + 3tz^2*G^2 + z^3*G^3.
Row n has 1+floor(n/2) terms.
Row sums yield A001764.
T(n,0) = A120985(n).
Sum_{k>=1} k*T(n,k) = 3*binomial(3n,n-2) = 3*A003408(n-2).

A120429 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k leaves (i.e., vertices of degree 0; n>=0, k>=1). A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

Original entry on oeis.org

1, 3, 9, 3, 27, 27, 1, 81, 162, 30, 243, 810, 360, 15, 729, 3645, 2970, 405, 3, 2187, 15309, 19845, 5670, 252, 6561, 61236, 115668, 56700, 6426, 84, 19683, 236196, 612360, 459270, 98658, 4536, 12, 59049, 885735, 3018060, 3214890, 1122660
Offset: 0

Views

Author

Emeric Deutsch, Jul 21 2006

Keywords

Comments

Row n has n + 1 - ceiling(n/3) terms.
Row sums yield A001764.
T(n,1) = 3^n = A000244(n).
Sum_{k>=1} k*T(n,k) = binomial(3n,n) = A005809(n).

Examples

			T(2,2)=3 because we have (Q,L,M), (Q,L,R) and (Q,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
    1;
    3;
    9,   3;
   27,  27,   1;
   81, 162,  30;
  243, 810, 360,  15;

Crossrefs

Cf. A001764, A000244, A005809, A120981, A120982, A120983.

Programs

  • Maple
    T:=proc(n,k) if k<=n+1-ceil(n/3) then (1/(n+1))*binomial(n+1,k)*sum(3^(n+j-2*k+2)*binomial(n+1-k,j)*binomial(j,k-1-j),j=0..n+1-k) else 0 fi end: 1; for n from 1 to 11 do seq(T(n,k),k=1..n+1-ceil(n/3)) od; # yields sequence in triangular form

Formula

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k}3^(n-2k+j+2)*binomial(n+1-k,j)*binomial(j,k-1-j).
G.f. = G = G(t,z) satisfies G = (1+z(G-1+t))^3.
Showing 1-3 of 3 results.

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