%I #15 Feb 21 2017 10:14:48

%S 0,1,1,0,0,1,1,0,0,1,1,0,0,2,2,0,1,3,3,1,2,6,6,3,6,11,11,9,15,23,24,

%T 25,39,48,52,67,96,107,122,174,242,247,295,448,598,598,744,1141,1493,

%U 1493,1913,2898,3730,3826,5003,7362,9396,9980,13201,18757,23840

%N Number of unlabeled rooted trees with n nodes where the outdegrees (branching factors) of adjacent nodes differ by exactly one.

%C These trees are also counted by A260353 and A260403.

%H Alois P. Heinz, <a href="/A257654/b257654.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%e a(5) = 1: o

%e . / \

%e . o o

%e . | |

%e . o o

%p b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),

%p `if`(i<1 or v<1 or n<v, 0, add(binomial(A(i, h)+j-1, j)*

%p b(n-i*j, i-1, h, v-j), j=0..min(n/i, v))))

%p end:

%p A:= proc(n, k) option remember; `if`(n=0, 0, add(`if`(j=k, 0,

%p b(n-1$2, j$2)), j=max(k-1, 0)..min(k+1, n-1)))

%p end:

%p a:= n-> add(b(n-1$2, j$2), j=0..n-1):

%p seq(a(n), n=0..60);

%t 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[If[j==k, 0, 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, 60}] (* _Jean-François Alcover_, Feb 21 2017, translated from Maple *)

%Y Cf. A000081, A259863, A260353, A260403, A253244.

%K nonn

%O 0,14

%A _Alois P. Heinz_, Jul 25 2015