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)
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)
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
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)
-
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
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)
-
\\ 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.