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 1-2 of 2 results.

A384685 Triangle read by rows: T(n,k) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, all internal nodes have weight 1, and leaf nodes have weights in {1,...,k}.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 5, 8, 9, 0, 14, 25, 28, 29, 0, 42, 83, 95, 98, 99, 0, 132, 289, 337, 349, 352, 353, 0, 429, 1041, 1236, 1285, 1297, 1300, 1301, 0, 1430, 3847, 4652, 4854, 4903, 4915, 4918, 4919, 0, 4862, 14504, 17865, 18709, 18912, 18961, 18973, 18976, 18977
Offset: 0

Views

Author

John Tyler Rascoe, Jun 06 2025

Keywords

Examples

			Triangle begins:
    k=0     1     2     3     4     5     6     7      8
 n=0 [1]
 n=1 [0,    1]
 n=2 [0,    2,    3]
 n=3 [0,    5,    8,    9]
 n=4 [0,   14,   25,   28,   29]
 n=5 [0,   42,   83,   95,   98,   99]
 n=6 [0,  132,  289,  337,  349,  352,  353]
 n=7 [0,  429, 1041, 1236, 1285, 1297, 1300, 1301]
 n=8 [0, 1430, 3847, 4652, 4854, 4903, 4915, 4918, 4919]
...
T(2,2) = 3 counts:
  o    o      o
  |    |     / \
 (2)  (1)  (1) (1)
       |
      (1)
		

Crossrefs

Cf. (column k=1) A000108, A078481, A078482, A088218, (column k=2) A143330, A380761, A384613.

Programs

  • PARI
    b(k) = {(x^2-x^(k+1))/(1-x)}
    P(N,k) = {my(x='x+O('x^N)); Vec((1-b(k)-sqrt((b(k)-1)^2-4*x))/(2*x))}
    T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1,1,0))~); for(k=1,N, v=matconcat([v,P(N+1,k)~])); vector(N,n, vector(n,k,v[n,k]))}

Formula

G.f. of column k is (1 - b(k,x) - sqrt((b(k,x) - 1)^2 - 4*x))/(2*x) where b(k,x) = (x^2 - x^(k + 1))/(1 - x).
T(n,k) = T(n,n) for k > n.

A382060 Number of rooted ordered trees with n nodes such that the degree of each node is less than or equal to its depth plus one.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 10, 27, 77, 231, 719, 2302, 7541, 25177, 85405, 293635, 1021272, 3587674, 12713796, 45402113, 163244197, 590529759, 2147915920, 7851127319, 28826079193, 106268313333, 393218951710, 1459969448090, 5437679646441, 20311366912839, 76072367645347, 285623120079865, 1074888308119285
Offset: 0

Views

Author

John Tyler Rascoe, Mar 14 2025

Keywords

Comments

The root vertex is depth d=0 and is to have <= d+1 = 1 children so these are "planted" trees.

Examples

			a(5) = 4 counts:
                               depth:
   o     o       o       o       0
   |     |       |       |
   o     o       o       o       1
   |     |      / \     / \
   o     o     o   o   o   o     2
   |    / \    |           |
   o   o   o   o           o     3
   |
   o                             4
		

Crossrefs

Programs

  • PARI
    \\ see links
  • Python
    # see links
    
Showing 1-2 of 2 results.