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%I #9 Feb 21 2017 10:23:41
%S 0,1,1,1,1,2,3,5,8,14,24,43,76,138,250,460,848,1576,2939,5516,10382,
%T 19629,37221,70820,135097,258426,495460,952083,1833176,3536502,
%U 6834408,13229829,25649202,49799891,96821854,188485968,367375883,716872030,1400373358
%N Number of unlabeled rooted trees with n nodes where the outdegrees (branching factors) of adjacent nodes differ by at most one.
%H Alois P. Heinz, <a href="/A260403/b260403.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) = 2:
%e : o o
%e : | / \
%e : o o o
%e : | | |
%e : o o o
%e : |
%e : o
%e : |
%e : 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,
%p add(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..40);
%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[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 *)
%Y Cf. A000081, A259863, A260353, A257654, A253244.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Jul 24 2015
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