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A244455 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 1.

%I A244455 #13 Feb 06 2015 08:11:42
%S A244455 1,1,3,7,17,42,105,267,684,1775,4639,12238,32491,86859,233496,631082,
%T A244455 1713613,4673455,12795426,35159212,96927479,268021520,743188706,
%U A244455 2066071045,5757360011,16079027344,44997313684,126166307275,354384737204,997083779801,2809751278062
%N A244455 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 1.
%H A244455 Alois P. Heinz, <a href="/A244455/b244455.txt">Table of n, a(n) for n = 2..800</a>
%F A244455 a(n) = A000081(n) - A001678(n+1).
%e A244455 a(5) = 7:
%e A244455   o    o      o      o      o      o      o
%e A244455   |    |      |     / \    / \     |     /|\
%e A244455   o    o      o    o   o  o   o    o    o o o
%e A244455   |    |     / \   |      |   |   /|\   |
%e A244455   o    o    o   o  o      o   o  o o o  o
%e A244455   |   / \   |      |
%e A244455   o  o   o  o      o
%e A244455   |
%e A244455   o
%p A244455 b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
%p A244455       1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
%p A244455       b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
%p A244455     end:
%p A244455 a:= n-> b(n-1$2, 1$2) -b(n-1$2, 2$2):
%p A244455 seq(a(n), n=2..35);
%t A244455 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 A244455 Column k=1 of A244454.
%Y A244455 Cf. A106640 (the same for ordered rooted trees).
%K A244455 nonn
%O A244455 2,3
%A A244455 _Joerg Arndt_ and _Alois P. Heinz_, Jun 29 2014