A317580 Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.
1, 1, 1, 3, 5, 12, 28, 66, 153, 367, 880, 2121, 5127, 12441, 30248, 73746, 180077, 440571, 1079438, 2648511, 6506170, 16001256, 39393173, 97074140, 239419963, 590972968, 1459808862, 3608483107, 8925476591, 22090139751, 54702648393, 135533335933, 335967782916
Offset: 1
Keywords
Examples
The a(6) = 12 rooted identity trees with a distinguished leaf: (((((O))))), (((O(o)))), (((o(O)))), ((O((o)))), ((o((O)))), (O(((o)))), (o(((O)))), ((O)((o))), ((o)((O))), (O(o(o))), (o(O(o))), (o(o(O))).
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..200
Crossrefs
Programs
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Mathematica
urit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[urit/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}]; Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urit[n]}],{n,10}] -
PARI
WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)} seq(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p -> subst(deriv(p), y, 1), v)} \\ Andrew Howroyd, Aug 28 2018
Formula
a(n) = Sum_{k=1, n} k*A055327(n, k). - Andrew Howroyd, Aug 28 2018
Extensions
Terms a(26) and beyond from Andrew Howroyd, Aug 28 2018
Comments