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
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)
-
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)
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
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)
-
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]))}
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
a(3) = 6 counts:
o o o o o __o__
| | | / \ / \ / | \
(3) (2) (1) (1) (2) (2) (1) (1) (1) (1)
| |
(1) (2)
-
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)
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
a(3) = 6 counts:
o o o o o __o__
| | | / \ / \ / | \
(3) (2) (1) (1) (2) (2) (1) (1) (1) (1)
| |
(1) (2)
-
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)
Showing 1-4 of 4 results.