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A363272 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where some child tree has k leaves, 1 <= k <= n/2.

%I A363272 #16 Jan 01 2024 19:51:25
%S A363272 1,1,1,1,2,1,3,2,1,6,3,2,11,6,3,3,23,11,6,6,46,23,11,12,6,98,46,23,22,
%T A363272 18,207,98,46,46,33,21,451,207,98,92,69,66,983,451,207,196,138,138,66,
%U A363272 2179,983,451,414,294,276,253,4850,2179,983,902,621,588,506,276
%N A363272 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where some child tree has k leaves, 1 <= k <= n/2.
%H A363272 Andrew Howroyd, <a href="/A363272/b363272.txt">Table of n, a(n) for n = 2..2501</a> (rows 2..100)
%F A363272 T(n,k) = A001190(k) * A001190(n-k) if k < n/2; otherwise
%F A363272 T(2k,k) = A001190(k) * (A001190(k) + 1) / 2 = A000217(A001190(n)).
%F A363272 Sum_{k >= 1} T(n,k) = A001190(n).
%F A363272 Sum_{i >= k} T(n,i) = A363273(n,k).
%F A363272 Sum_{i <= n-1, i+j >= n} T(i,j) = A000671(n-2).
%e A363272 Table begins:
%e A363272  1;
%e A363272  1;
%e A363272  1,  1;
%e A363272  2,  1;
%e A363272  3,  2,  1;
%e A363272  6,  3,  2;
%e A363272 11,  6,  3,  3;
%e A363272 23, 11,  6,  6;
%e A363272 46, 23, 11, 12,  6;
%e A363272 98, 46, 23, 22, 18;
%e A363272 ...
%o A363272 (PARI)
%o A363272 T(n)={my(A=vector(n), R=vector(n)); A[1]=1; R[1]=[]; for(i=2, n, R[i] = vector(i\2, j, if(2*j<i, A[j] * A[i-j], A[i/2] * (A[i/2] + 1)/2)); A[i] = vecsum(R[i])); R}
%o A363272 { my(A=T(12)); for(n=2, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 01 2024
%Y A363272 Row sums are A001190.
%Y A363272 First column k=1 is T(n,1) = A001190(n-1).
%Y A363272 Cf. A000671, A363273.
%K A363272 nonn,tabf
%O A363272 2,5
%A A363272 _Harry Richman_, May 24 2023
%E A363272 Terms a(32) and beyond from _Andrew Howroyd_, Jan 01 2024