Original entry on oeis.org
0, 1, 4, 20, 115, 719, 4766, 32973, 235381, 1721159, 12826228, 97055181, 743724984, 5759636510, 45007066269, 354426847597, 2809934352700, 22409533673568, 179655930440464, 1447023384581029, 11703780079612453, 95020085893954917, 774088023431472074
Offset: 0
-
with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<2, n,
(add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
end:
a:= n-> b(2*n):
seq(a(n), n=0..50); # Alois P. Heinz, May 16 2013
-
b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]; a[n_] := b[2*n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
A299039
Number of rooted trees with 2n nodes where each node has at most n children.
Original entry on oeis.org
1, 1, 3, 17, 106, 693, 4690, 32754, 234746, 1719325, 12820920, 97039824, 743680508, 5759507657, 45006692668, 354425763797, 2809931206626, 22409524536076, 179655903886571, 1447023307374888, 11703779855021636, 95020085240320710, 774088021528328920
Offset: 0
a(2) = 3:
o o o
| | / \
o o o o
| / \ |
o o o o
|
o
-
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
end:
a:= n-> `if`(n=0, 1, b(2*n-1$2, n$2)):
seq(a(n), n=0..25);
-
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[2n - 1, 2n - 1, n, n]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, from Maple *)
A299113
Number of rooted identity trees with 2n+1 nodes.
Original entry on oeis.org
1, 1, 3, 12, 52, 247, 1226, 6299, 33209, 178618, 976296, 5407384, 30283120, 171196956, 975662480, 5599508648, 32334837886, 187737500013, 1095295264857, 6417886638389, 37752602033079, 222861754454841, 1319834477009635, 7839314017612273, 46688045740233741
Offset: 0
a(2) = 3:
o o o
| | / \
o o o o
| / \ |
o o o o
| | |
o o o
|
o
-
with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(
b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
end:
a:= n-> b(2*n+1):
seq(a(n), n=0..30);
-
b[n_] := b[n] = If[n < 2, n, Sum[b[n - k]*Sum[b[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1)];
a[n_] := b[2*n + 1];
Array[a, 30, 0] (* Jean-François Alcover, May 30 2019, from Maple *)
-
from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def b(n): return n if n<2 else sum([b(n-k)*sum([b(d)*d*(-1)**(k//d+1) for d in divisors(k)]) for k in range(1, n)])//(n-1)
def a(n): return b(2*n+1)
print([a(n) for n in range(31)]) # Indranil Ghosh, Mar 02 2018
Showing 1-3 of 3 results.