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A360985 Triangle read by rows: T(n,k) is the number of full binary trees with n leaves, each internal node having the heights of its two subtrees weakly increasing left to right, and with k internal nodes having two subtrees of equal height.

%I A360985 #11 Apr 03 2023 21:47:36
%S A360985 1,0,1,0,1,0,0,1,0,1,0,1,1,1,0,0,1,2,2,1,0,0,1,4,3,2,2,0,0,1,6,7,6,3,
%T A360985 0,1,0,1,9,13,14,9,3,1,0,0,1,12,27,27,22,14,3,1,0,0,1,16,47,59,54,32,
%U A360985 16,7,0,0,0,1,20,81,117,125,91,44,20,8,1,0
%N A360985 Triangle read by rows: T(n,k) is the number of full binary trees with n leaves, each internal node having the heights of its two subtrees weakly increasing left to right, and with k internal nodes having two subtrees of equal height.
%H A360985 Andrew Howroyd, <a href="/A360985/b360985.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50).
%F A360985 T(n,2) = A002620(n-3) for all n>=3.
%F A360985 T(n,n-1) = 1 if n is a power of 2, and T(n,n) = 0 otherwise.
%F A360985 T(n,n-2) != 0 if and only if n-1 has exactly one maximal group of consecutive zeros in its binary representation, and in this case T(n,n-2) = 2^(a-1) where a is the number of 1s at the beginning of the binary representation of n-1.
%F A360985 Sum_{k=0..n-1} T(n,k)*2^(n-k-1) = A000108(n-1).
%e A360985 The table for T(n,k) begins:
%e A360985   n\k 0   1   2   3    4    5    6    7    8   9  10  11  12  13  14  15
%e A360985    1  1
%e A360985    2  0   1
%e A360985    3  0   1   0
%e A360985    4  0   1   0   1
%e A360985    5  0   1   1   1    0
%e A360985    6  0   1   2   2    1    0
%e A360985    7  0   1   4   3    2    2    0
%e A360985    8  0   1   6   7    6    3    0    1
%e A360985    9  0   1   9  13   14    9    3    1    0
%e A360985   10  0   1  12  27   27   22   14    3    1   0
%e A360985   11  0   1  16  47   59   54   32   16    7   0   0
%e A360985   12  0   1  20  81  117  125   91   44   20   8   1   0
%e A360985   13  0   1  25 128  233  272  228  143   61  23   8   2   0
%e A360985   14  0   1  30 197  439  573  555  389  206  90  21  10   2   0
%e A360985   15  0   1  36 287  801 1178 1275 1014  621 303 109  32   4   4   0
%e A360985   16  0   1  42 410 1383 2367 2841 2522 1727 962 421 138  36   7   0   1
%o A360985 (PARI) T(n)={my(p=x+O(x*x^n), q=p); for(n=2, n, p=y*p^2 + p*(q-p); q+=p); my(v=Vec(q)); vector(#v, n, Vecrev(v[n], n))}
%o A360985 { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Mar 24 2023
%Y A360985 Row sums give A045761.
%Y A360985 Column k=2 is A002620.
%Y A360985 Cf. A000108, A335833.
%K A360985 nonn,tabl
%O A360985 1,18
%A A360985 _Ludovic Schwob_, Feb 27 2023