cp's OEIS Frontend

A301422 Regular triangle where T(n,k) is the number of r-trees of size n with k leaves.

%I A301422 #9 Aug 26 2018 16:37:17
%S A301422 1,1,0,1,1,0,1,2,1,0,1,4,3,1,0,1,6,8,4,1,0,1,9,19,14,5,1,0,1,12,36,40,
%T A301422 21,6,1,0,1,16,65,102,75,30,7,1,0,1,20,106,223,224,123,40,8,1,0,1,25,
%U A301422 168,457,604,439,191,52,9,1,0,1,30,248,847,1433,1346,764,276
%N A301422 Regular triangle where T(n,k) is the number of r-trees of size n with k leaves.
%C A301422 An r-tree (A093637) of size n > 0 is a finite sequence of r-trees with weakly decreasing sizes summing to n - 1. This is a similar construction to p-trees (A196545) except that r-trees are not required to be series-reduced and are weighted by all nodes (including the root) rather than just the leaves.
%H A301422 Andrew Howroyd, <a href="/A301422/b301422.txt">Table of n, a(n) for n = 1..1275</a>
%e A301422 Triangle begins:
%e A301422   1
%e A301422   1   0
%e A301422   1   1   0
%e A301422   1   2   1   0
%e A301422   1   4   3   1   0
%e A301422   1   6   8   4   1   0
%e A301422   1   9  19  14   5   1   0
%e A301422   1  12  36  40  21   6   1   0
%e A301422   1  16  65 102  75  30   7   1   0
%e A301422   1  20 106 223 224 123  40   8   1   0
%e A301422   1  25 168 457 604 439 191  52   9   1   0
%e A301422   ...
%e A301422 The T(6,3) = 8 r-trees: (((ooo))), (((oo)o)), (((o)oo)), (((oo))o), (((o)o)o), ((oo)(o)), (((o))oo), ((o)(o)o).
%t A301422 rtrees[n_]:=Join@@Table[Tuples[rtrees/@y],{y,IntegerPartitions[n-1]}];
%t A301422 Table[Length[Select[rtrees[n],Count[#,{},{-2}]===k&]],{n,8},{k,n}]
%o A301422 (PARI) A(n)={my(v=vector(n)); v[1]=y; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); vector(n, k, Vecrev(v[k]/y,k))}
%o A301422 { my(T=A(10)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Aug 26 2018
%Y A301422 Cf. A000081, A003238, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A290689, A291443, A297791, A300443, A301342-A301345, A301364.
%K A301422 nonn,tabl
%O A301422 1,8
%A A301422 _Gus Wiseman_, Mar 20 2018