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A036602 Triangle of coefficients of generating function of binary rooted trees of height at most n.

%I A036602 #28 Feb 14 2021 13:02:51
%S A036602 1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,1,1,1,2,3,5,6,8,8,9,7,7,4,3,1,1,1,1,1,
%T A036602 2,3,6,10,17,25,38,52,73,93,121,143,172,187,205,202,201,177,158,123,
%U A036602 99,66,47,26,17,7,4,1,1,1,1,1,2,3,6,11,22,39,70,118,200,324,526
%N A036602 Triangle of coefficients of generating function of binary rooted trees of height at most n.
%H A036602 Alois P. Heinz, <a href="/A036602/b036602.txt">Rows n = 0..12, flattened</a>
%H A036602 E. M. Rains and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/cayley.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A036602 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%e A036602 Triangle begins:
%e A036602 1
%e A036602 1, 1;
%e A036602 1, 1, 1, 1;
%e A036602 1, 1, 1, 2, 2, 2,  1,  1;
%e A036602 1, 1, 1, 2, 3, 5,  6,  8,  8,  9,   7,   7,   4,    3,    1,    1;
%e A036602 1, 1, 1, 2, 3, 6, 10, 17, 25, 38,  52,  73,  93,  121,  143,  172,  187, ...
%e A036602 1, 1, 1, 2, 3, 6, 11, 22, 39, 70, 118, 200, 324,  526,  825, 1290, 1958, ...
%e A036602 1, 1, 1, 2, 3, 6, 11, 23, 45, 90, 171, 325, 598, 1097, 1972, 3531, 6225, ...
%p A036602 b:= proc(n, h) option remember; `if`(n<2, n, `if`(h<1, 0, `if`(n::odd, 0,
%p A036602      (t-> t*(1-t)/2)(b(n/2, h-1)))+add(b(i, h-1)*b(n-i, h-1), i=1..n/2)))
%p A036602     end:
%p A036602 A:= (n, k)-> b(k+1, n):
%p A036602 seq(seq(A(n, k), k=0..2^n-1), n=0..6);  # _Alois P. Heinz_, Sep 08 2017
%t A036602 b[n_, h_] := b[n, h] = If[n < 2, n, If[h < 1, 0, If[OddQ[n], 0, Function[t, t*(1-t)/2][b[n/2, h-1]]] + Sum[b[i, h-1]*b[n-i, h-1], {i, 1, n/2}]]];
%t A036602 A[n_, k_] := b[k+1, n];
%t A036602 Table[Table[A[n, k], {k, 0, 2^n-1}], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Feb 14 2021, after _Alois P. Heinz_ *)
%Y A036602 Cf. A001190, A036587, A036588, A036589, A036590, A036591, A036592.
%K A036602 nonn,tabf,nice,easy
%O A036602 0,11
%A A036602 _N. J. A. Sloane_