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.

A384747 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, non-root node weights are in {1,..,k}, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 6, 0, 1, 11, 15, 16, 0, 1, 26, 39, 43, 44, 0, 1, 63, 110, 123, 127, 128, 0, 1, 153, 308, 358, 371, 375, 376, 0, 1, 376, 869, 1046, 1096, 1109, 1113, 1114, 0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346, 0, 1, 2317, 7238, 9283, 9904, 10084, 10134, 10147, 10151, 10152
Offset: 0

Views

Author

John Tyler Rascoe, Jun 09 2025

Keywords

Examples

			Triangle begins:
    k=0  1    2     3     4     5     6     7     8     9
 n=0 [1]
 n=1 [0, 1]
 n=2 [0, 1,   2]
 n=3 [0, 1,   5,    6]
 n=4 [0, 1,  11,   15,   16]
 n=5 [0, 1,  26,   39,   43,   44]
 n=6 [0, 1,  63,  110,  123,  127,  128]
 n=7 [0, 1, 153,  308,  358,  371,  375,  376]
 n=8 [0, 1, 376,  869, 1046, 1096, 1109, 1113, 1114]
 n=9 [0, 1, 931, 2499, 3098, 3278, 3328, 3341, 3345, 3346]
...
T(3,3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Crossrefs

Cf. A051286 (column k=2), A382096 (column k=3), A384748 (main diagonal).
Cf. A000108, A002212, A143330, A384613, A384685.

Programs

  • PARI
    b(i,j,k,N) = {if(k>N,1, 1/( 1  - sum(u=1,j, if(u==i,0,x^u * b(u,j,k+1,N-u+1)))))}
Gx(k,N) = {my(x='x+O('x^(N+1))); Vec(1/(1 - sum(i=1,k, b(i,k,1,N)*x^i)))}
T(max_row) = { my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Gx(k,N)~])); vector(N, n, vector(n, k, v[n, k]))}
T(9)

Formula

T(n,k) = T(n,n) for k > n.

A384937 Number for rooted ordered trees with edge weights summing to n, where edge weights are all greater than zero, and the sequences of edge weights in all downward paths are weakly increasing.

Original entry on oeis.org

1, 1, 3, 9, 30, 103, 372, 1379, 5248, 20356, 80252, 320581, 1295018, 5280967, 21711163, 89890559, 374478935, 1568585095, 6602283315, 27910296899, 118448905668, 504466997897, 2155412350793, 9236401247438, 39686616306747, 170946789568804, 738024717474360
Offset: 0

Views

Author

John Tyler Rascoe, Jun 12 2025

Keywords

Examples

			The following tree with sum of edge weights 13 contains downward paths of edge weights (1), (2,3,4), and (2,3,3) all of which are weakly increasing. So this tree is counted under a(13) = 5280967.
           o
        2 / \ 1
         o   o
      3 / 	
       o
    4 / \ 3	
     o   o

Crossrefs

Cf. A000108, A000169, A000957, A001764, A036765, A384747, A384748.

Programs

  • PARI
    w(j,k,N) = {if(k>N,1, 1/(1 - sum(i=j,N, x^i * w(i,k+1,N-i+1))))}
Ax(N) = {Vec(w(1,1,N)+ O('x^(N+1)))}
Ax(10)

Formula

G.f.: G_1(x) where G_k(x) = 1/(1 - Sum_{i>=k} x^i * G_i(x)).

A384938 Number for rooted ordered trees with edge weights summing to n, where edge weights are all greater than zero, and the sequences of edge weights in all downward paths are strictly increasing.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 61, 142, 334, 785, 1845, 4339, 10211, 24030, 56560, 133143, 313433, 737906, 1737275, 4090206, 9630067, 22673482, 53383917, 125691264, 295938451, 696785116, 1640579144, 3862745470, 9094847357, 21413863699, 50419073794, 118712060012, 279508439419
Offset: 0

Views

Author

John Tyler Rascoe, Jun 13 2025

Keywords

Examples

			The following tree with sum of edge weights 15 contains downward paths of edge weights (1), (2,3,4), and (2,3,5) all of which are weakly increasing. So this tree is counted under a(13) = 133143.
           o
        2 / \ 1
         o   o
      3 / 	
       o
    4 / \ 5	
     o   o

Crossrefs

Cf. A000108, A000169, A000957, A001764, A036765, A384747, A384748.

Programs

  • PARI
    w(j,k,N) = {if(k>N,1, 1/(1 - sum(i=j+1,N, x^i * w(i,k+1,N-i+1))))}
Bx(N) = {my(x='x+O('x^(N+1))); Vec(w(0,1,N)+ O('x^(N+1)))}
Bx(10)

Formula

G.f.: G_0(x) where G_k(x) = 1/(1 - Sum_{i>k} x^i * G_i(x)).
Showing 1-3 of 3 results.

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