A304173
Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.
Original entry on oeis.org
1, 1, 2, 5, 13, 34, 90, 242, 660, 1822, 5085, 14333, 40759, 116817, 337140, 979098, 2859439, 8393113, 24747052, 73262246, 217681621, 648939319, 1940461444, 5818595438, 17492367097, 52712114792, 159193762250, 481754196170, 1460650624068, 4436422703787, 13496947320929
Offset: 1
The a(5) = 13 plane trees:
((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
(((o))o), (o((o))), (o(oo)), ((o)(o)),
((o)oo), (o(o)o), (oo(o)),
(oooo).
Missing from this list is ((oo)o).
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pplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[pplane/@c],OrderedQ[Count[#,{},{0,Infinity}]&/@#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[pplane[n]],{n,10}]
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seq(n)={my(p=x*y+O(x^2)); for(n=2, n, p=x*(y-1 + 1/prod(k=1, n-1, 1 - y^k*polcoef(p,k,y)))); Vec(subst(p,y,1))} \\ Andrew Howroyd, Jan 22 2021
A304175
Number of leaf-balanced rooted plane trees with n nodes.
Original entry on oeis.org
1, 1, 2, 5, 12, 27, 59, 128, 277, 597, 1280, 2730, 5794, 12248, 25836, 54508, 115222, 244144, 518104, 1099499, 2330326, 4930089, 10415135, 21992400, 46470911, 98353146, 208580686, 443186181, 942988423, 2007981801, 4276830431, 9109431322, 19404918449, 41357252072, 88236092543
Offset: 1
The a(5) = 12 leaf-balanced plane trees:
((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
(((o))o), (o((o))), ((o)(o)),
((o)oo), (o(o)o), (oo(o)),
(oooo).
Missing from this list are ((oo)o) and (o(oo)).
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lbplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lbplane/@c],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lbplane[n]],{n,10}]
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seq(n)={my(v=vector(n)); v[1]=x/(1-x) + O(x*x^n); for(k=2, n, v[k]=x*sumdiv(k, d, if(dAndrew Howroyd, Dec 13 2020
A298535
Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of leaves.
Original entry on oeis.org
1, 1, 1, 2, 5, 13, 32, 80, 200, 511, 1323, 3471, 9183, 24491, 65715, 177363, 481135, 1311340, 3589023, 9860254, 27181835, 75165194, 208439742, 579522977, 1615093755, 4511122964, 12625881944, 35405197065, 99459085125, 279861792874, 788712430532, 2226015529592
Offset: 1
Cf.
A000081,
A003238,
A004111,
A032305,
A289079,
A290689,
A291443,
A297791,
A298422,
A298533,
A298536.
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rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
Table[Length[Select[rut[n],UnsameQ@@(Count[#,{},{0,Infinity}]&/@#)&]],{n,15}]
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\\ here R is A055277 as vector of polynomials
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1 + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
seq(n) = {my(M=Mat(apply(p->Colrev(p,n), R(n-1)))); Vec(prod(i=2, #M, 1 + x*Ser(M[i,])))} \\ Andrew Howroyd, May 20 2018
A298537
Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.
Original entry on oeis.org
1, 1, 2, 3, 6, 10, 25, 49, 127, 291, 766, 1843, 5003, 12487, 34151, 87983, 242088, 634848, 1763749, 4688677, 13085621, 35241441, 98752586, 268282856, 755353825, 2067175933, 5837592853, 16087674276, 45550942142, 126186554309, 358344530763, 997171512999
Offset: 1
The a(5) = 6 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), ((o)(o)), (oooo).
Cf.
A000081,
A003238,
A004111,
A032305,
A289078,
A289079,
A290689,
A291443,
A297791,
A298422,
A298533,
A298535,
A298538,
A298539.
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r[n_]:=r[n]=If[n===1,1,Sum[Product[Binomial[r[x]+Count[ptn,x]-1,Count[ptn,x]],{x,Union[ptn]}],{ptn,IntegerPartitions[n-1]}]];
Table[If[n===1,1,Sum[Binomial[r[(n-1)/d]+d-1,d],{d,Divisors[n-1]}]],{n,40}]
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