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A298478 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are different.

%I A298478 #15 Feb 02 2021 18:20:53
%S A298478 1,1,1,3,3,5,13,15,23,34,95,106,176,241,374,942,1129,1760,2515,3711,
%T A298478 5136,12857,14911,23814,33002,49141,65798,97056,209707,255042,389725,
%U A298478 545290,790344,1071010,1525919,2043953,4272124,5110583,7772247,10611491,15447864,20496809
%N A298478 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are different.
%C A298478 a(n) is the number of labeled trees with sum of the labels equal to n-1 and the outdegree of every node less than or equal to the value of its label. - _Andrew Howroyd_, Feb 02 2021
%H A298478 Andrew Howroyd, <a href="/A298478/b298478.txt">Table of n, a(n) for n = 1..100</a>
%e A298478 The a(7) = 13 trees: ((o(ooo))), ((oo(oo))), ((ooooo)), (o((ooo))), (o(oo(o))), (o(oooo)), ((o)(ooo)), (oo((oo))), (oo(o(o))), (o(o)(oo)), (ooo(oo)), (oooo(o)), (oooooo).
%t A298478 krut[n_]:=krut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[krut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@Length/@Cases[#,{__},{0,Infinity}]&]];
%t A298478 Table[krut[n]//Length,{n,15}]
%o A298478 (PARI)
%o A298478 relabel(b)={my(w=hammingweight(b)); b = bitand((1<<w)-1, b); b + (((1 << (w-hammingweight(b))) - 1) << w)}
%o A298478 a(n)={local(M=Map());
%o A298478   my(recurse(b,k) = if(!b, 1, b=relabel(b); my(hk=[b,k], z); if(!mapisdefined(M, hk, &z),
%o A298478     z = if(k==0,
%o A298478       sum(i=1, logint(b,2), if(bittest(b,i), self()(b-2^i, i-1))),
%o A298478       sum(f=2^logint(b,2), b, if(!bitnegimply(f,b), self()(f,0)*self()(b-f,k-1)));
%o A298478     );
%o A298478     mapput(M, hk, z)); z));
%o A298478   if( n==1, 1, my(t=0); for(np=1, sqrtint(2*n-2), forpart(p=n-1-binomial(np,2), t+=recurse(sum(i=1, #p, 2^(p[i]+i-1)), 0), , [np,np]));  t);
%o A298478 } \\ _Andrew Howroyd_, Feb 02 2021
%Y A298478 Cf. A000081, A001190, A001678, A004111, A032305, A124343, A290689, A295461, A298118, A298304, A298422, A298479.
%K A298478 nonn
%O A298478 1,4
%A A298478 _Gus Wiseman_, Jan 19 2018
%E A298478 a(27)-a(34) from _Robert G. Wilson v_, Jan 19 2018
%E A298478 Terms a(35) and beyond from _Andrew Howroyd_, Feb 02 2021