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A072638 Number of unary-binary rooted trees of height at most n.

%I A072638 #25 Aug 12 2017 09:57:54
%S A072638 0,1,3,10,66,2278,2598060,3374961778891,5695183504492614029263278,
%T A072638 16217557574922386301420536972254869595782763547560
%N A072638 Number of unary-binary rooted trees of height at most n.
%C A072638 A unary-binary tree is one in which the degree of every node is <= 3.
%C A072638 a(n+1) = (a(n)+1)-th triangular numbers = A000217(a(n)+1). a(n+1) = (a(n) + 1) * (a(n) + 2) / 2. a(n+1) = A006894(n+2) - 1. - _Jaroslav Krizek_, Sep 11 2009
%C A072638 a(n) is the smallest integer that is the sum of n distinct members of the complete sequence A000124. See A204009 for the binary vectors that select the terms from A000124. - _Frank M Jackson_, Jan 09 2012
%H A072638 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A072638 a(n+1) = 1 + (a(n)*(a(n)+3))/2.
%F A072638 Conjecture: a(n) = A006894(n+1) - 1. - _R. J. Mathar_, Apr 23 2007
%F A072638 a(n) := C(a(n-1) + 2, 2), n >= -1. - _Zerinvary Lajos_, Jun 08 2007
%p A072638 a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n], n=-1..9); # _Zerinvary Lajos_, Jun 08 2007
%t A072638 Clear[a]; a[0] = 0; a[n_] := a[n] = 1 + (a[n-1]*(a[n-1]+3))/2; Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, Jan 31 2013 *)
%Y A072638 Maximal position in A071673 where the value n occurs.
%Y A072638 Binary width of each term: A072641. Cf. A072639, A072640, A072654.
%K A072638 nonn
%O A072638 0,3
%A A072638 _Antti Karttunen_, Jun 02 2002
%E A072638 Edited by _Christian G. Bower_, Oct 23 2002