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A319590 Number of binary rooted trees with n leaves spanning an initial interval of positive integers and all non-leaf nodes having out-degree 2.

%I A319590 #11 Apr 10 2020 06:16:14
%S A319590 1,2,8,58,576,7440,117628,2201014,47552012,1164812674,31898271660,
%T A319590 965666303078,32022547868872,1154362247246714,44945574393963472,
%U A319590 1879720975031634318,84039891496643620196,3999886612000379135606,201919706444252727224852,10775953237291840618917900
%N A319590 Number of binary rooted trees with n leaves spanning an initial interval of positive integers and all non-leaf nodes having out-degree 2.
%H A319590 Andrew Howroyd, <a href="/A319590/b319590.txt">Table of n, a(n) for n = 1..200</a>
%p A319590 b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
%p A319590       (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
%p A319590     end:
%p A319590 a:= n-> add(add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k), k=0..n):
%p A319590 seq(a(n), n=1..23);  # _Alois P. Heinz_, Sep 07 2019
%t A319590 b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1 - t)/2][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
%t A319590 a[n_] := Sum[Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}], {k, 0, n}];
%t A319590 Array[a, 23] (* _Jean-François Alcover_, Apr 10 2020, after _Alois P. Heinz_ *)
%o A319590 (PARI) \\ here R(n, k) is k-th column of A319539 as a vector.
%o A319590 R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
%o A319590 seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )}
%Y A319590 Row sums of A319541.
%Y A319590 Cf. A316651, A319539.
%K A319590 nonn
%O A319590 1,2
%A A319590 _Andrew Howroyd_, Sep 23 2018