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 1-5 of 5 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.

A384748 Number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are greater than 0, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 1, 2, 6, 16, 44, 128, 376, 1114, 3346, 10152, 31028, 95474, 295532, 919446, 2873388, 9015812, 28390466, 89689586, 284173096, 902780060, 2875016084, 9176388532, 29349499212, 94050228650, 301918397716, 970815092346
Offset: 0

Views

Author

John Tyler Rascoe, Jun 09 2025

Keywords

Examples

			a(3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Crossrefs

Cf. A000108, A002212, A143330, A384613, A384685, (main diagonal of A384747).

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)))))}
Dx(N) = {my(x='x+O('x^(N+1))); Vec(1/(1 - sum(i=1,N, b(i,N,1,N)*x^i)))}
Dx(10)

Extensions

a(14)-a(26) from David Radcliffe, Jun 10 2025

A382096 Number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2,3}, and no nodes have the same weight as their parent node.

Original entry on oeis.org

1, 1, 2, 6, 15, 39, 110, 308, 869, 2499, 7238, 21086, 61871, 182523, 540830, 1609238, 4805871, 14398559, 43264896, 130347450, 393650751, 1191441349, 3613345360, 10978726634, 33414836743, 101863289331, 310984519412, 950734751040, 2910319385881, 8919643999157, 27368321239074
Offset: 0

Views

Author

John Tyler Rascoe, Jun 08 2025

Keywords

Examples

			a(3) = 6 counts:
  o    o    o      o        o        __o__
  |    |    |     / \      / \      /  |  \
 (3)  (2)  (1)  (1) (2)  (2) (1)  (1) (1) (1)
       |    |
      (1)  (2)

Crossrefs

Cf. A000108, A002212, A143330, A384613, A384685, (column k=3 of A384747).

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)))}
Gx(3,20)

Formula

G.f.: G(x) = 1/(1 - b_1(x)*x - b_2(x)*x^2 - b_3(x)*x^3) where b_1(x) = 1/(1 - b_2(x)*x^2 - b_3(x)*x^3), b_2(x) = 1/(1 - b_1(x)*x - b_3(x)*x^3), b_3(x) = 1/(1 - b_1(x)*x - b_2(x)*x^2).

A385123 Triangle Read by rows: T(n,k) is the number of rooted ordered trees with n non-root nodes with non-root node labels in {1,..,k} such that all labels appear at least once in all groups of sibling nodes.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 5, 6, 6, 0, 14, 22, 36, 24, 0, 42, 90, 150, 240, 120, 0, 132, 378, 648, 1560, 1800, 720, 0, 429, 1638, 3318, 8400, 16800, 15120, 5040, 0, 1430, 7278, 18180, 43128, 126000, 191520, 141120, 40320, 0, 4862, 32946, 98502, 238320, 834120, 1905120, 2328480, 1451520, 362880
Offset: 0

Views

Author

John Tyler Rascoe, Jun 18 2025

Keywords

Examples

			Triangle begins:
    k=0    1    2      3     4      5      6     7
 n=0 [1]
 n=1 [0,   1]
 n=2 [0,   2,   2]
 n=3 [0,   5,   6,     6]
 n=4 [0,  14,  22,    36,   24]
 n=5 [0,  42,  90,   150,  240,   120]
 n=6 [0, 132,  378,  648, 1560,  1800,   720]
 n=7 [0, 429, 1638, 3318, 8400, 16800, 15120, 5040]
...
T(3,2) = 6 counts the three leaf permutations of each of the following trees:
      __o__        __o__
     /  |  \      /  |  \
   (1) (1) (2)  (1) (2) (2)

Crossrefs

Cf. A000108 (column k=1), A000142 (main diagonal), A385125 (row sums).
Cf. A107429, A384685, A384747.

Programs

  • PARI
    subsets(S) = {my(s=List()); for(i=0, 2^(#S) -1, my(x=List()); for(j=1,#S, if(bitand(i, 1<<(j-1)), listput(x, S[j]))); listput(s,Vec(x))); Vec(s)}
C_aB(B) = {my(S = subsets(B)); sum(i=1,#S, (1/(1-x*z*#S[i]))*(-1)^(#B-#S[i]))}
D(k,N,B) = {if(k>N,1, substpol(C_aB(B),z,1 + D(k+1,N-#B+1,B)))}
Dx(N,B) = {Vec(1+D(1,N,B)+ O('x^(N+1)))}
T(max_row) = {my( N = max_row+1, v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Dx(N+1, vector(k,i,i))~])); vector(N, n, vector(n, k, v[n, k]))}
T(8)

A385125 Number of rooted ordered trees with n non-root nodes all labeled with numbers greater than 0 such that the labels of all groups of sibling nodes cover the same initial interval.

Original entry on oeis.org

1, 1, 4, 17, 96, 642, 5238, 50745, 568976, 7256750, 103622742, 1634819518, 28208152974, 528060735100, 10654676857578
Offset: 0

Views

Author

John Tyler Rascoe, Jun 18 2025

Keywords

Examples

			Tree A has sibling node groups whose labels both cover the initial interval (1,2). Tree B has sibling node groups whose labels cover the initial intervals (1,2) and (1,2,3). So tree A is counted under a(5) = 642 while tree B is not.
  A:    __o__     B:    __o__
       /  |  \         /  |  \
     (1) (1) (2)     (3) (1) (2)
     / \             / \
   (1) (2)         (1) (2)

Crossrefs

Cf. Row sums of A385123.
Cf. A000108, A107429, A384685, A384747.

Programs

  • PARI
    \\ See A385123 for Dx(N,B)
Ax(N) = {my( v = vector(N, i, if(i==1, 1, 0))~); for(k=1, N, v=matconcat([v, Dx(N+1, vector(k,i,i))~])); vector(N, n, sum(i=1, n, v[n, i]))}
Ax(5)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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