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Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 1.
3

%I #13 Feb 06 2015 08:11:42

%S 1,1,3,7,17,42,105,267,684,1775,4639,12238,32491,86859,233496,631082,

%T 1713613,4673455,12795426,35159212,96927479,268021520,743188706,

%U 2066071045,5757360011,16079027344,44997313684,126166307275,354384737204,997083779801,2809751278062

%N Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 1.

%H Alois P. Heinz, <a href="/A244455/b244455.txt">Table of n, a(n) for n = 2..800</a>

%F a(n) = A000081(n) - A001678(n+1).

%e a(5) = 7:

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%p b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],

%p 1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*

%p b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))

%p end:

%p a:= n-> b(n-1$2, 1$2) -b(n-1$2, 2$2):

%p seq(a(n), n=2..35);

%t b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 1, 1] - b[n - 1, n - 1, 2, 2]; Table[a[n], {n, 2, 35}] (* _Jean-François Alcover_, Feb 06 2015, after Maple *)

%Y Column k=1 of A244454.

%Y Cf. A106640 (the same for ordered rooted trees).

%K nonn

%O 2,3

%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 29 2014