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A260403
Number of unlabeled rooted trees with n nodes where the outdegrees (branching factors) of adjacent nodes differ by at most one.
4
0, 1, 1, 1, 1, 2, 3, 5, 8, 14, 24, 43, 76, 138, 250, 460, 848, 1576, 2939, 5516, 10382, 19629, 37221, 70820, 135097, 258426, 495460, 952083, 1833176, 3536502, 6834408, 13229829, 25649202, 49799891, 96821854, 188485968, 367375883, 716872030, 1400373358
OFFSET
0,6
EXAMPLE
a(5) = 2:
: o o
: | / \
: o o o
: | | |
: o o o
: |
: o
: |
: o
MAPLE
b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
`if`(i<1 or v<1 or n<v, 0, add(binomial(A(i, h)+j-1, j)*
b(n-i*j, i-1, h, v-j), j=0..min(n/i, v))))
end:
A:= proc(n, k) option remember; `if`(n=0, 0,
add(b(n-1$2, j$2), j=max(k-1, 0)..min(k+1, n-1)))
end:
a:= n-> add(b(n-1$2, j$2), j=0..n-1):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, i_, h_, v_] := b[n, i, h, v] = If[n==0, If[v==0, 1, 0], If[i<1 || v<1 || n<v, 0, Sum[Binomial[A[i, h]+j-1, j]*b[n-i*j, i-1, h, v-j], {j, 0, Min[n/i, v]}]]]; A[n_, k_] := A[n, k] = If[n==0, 0, Sum[b[n-1, n-1, j, j], {j, Max[k-1, 0], Min[k+1, n-1]}]]; a[n_] := Sum[b[n-1, n-1, j, j], {j, 0, n-1}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 21 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 24 2015
STATUS
approved