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A363273 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where both children have at least k leaves, 1 <= k <= n/2.

%I A363273 #17 Jan 01 2024 19:46:27
%S A363273 1,1,2,1,3,1,6,3,1,11,5,2,23,12,6,3,46,23,12,6,98,52,29,18,6,207,109,
%T A363273 63,40,18,451,244,146,100,54,21,983,532,325,227,135,66,2179,1196,745,
%U A363273 538,342,204,66,4850,2671,1688,1237,823,529,253,10905,6055,3876,2893,1991,1370,782,276
%N A363273 Irregular triangle read by rows: T(n,k) = number of unlabeled binary rooted trees with n leaves, where both children have at least k leaves, 1 <= k <= n/2.
%H A363273 Andrew Howroyd, <a href="/A363273/b363273.txt">Table of n, a(n) for n = 2..2501</a> (rows 1..100)
%F A363273 T(n,k) = Sum_{j >= k} A363272(n,j).
%F A363273 Sum_{k >= 1} T(n-k, k) = A000671(n-2).
%e A363273 Table begins:
%e A363273   1;
%e A363273   1;
%e A363273   2,   1;
%e A363273   3,   1;
%e A363273   6,   3,  1;
%e A363273  11,   5,  2;
%e A363273  23,  12,  6,  3;
%e A363273  46,  23, 12,  6;
%e A363273  98,  52, 29, 18,  6;
%e A363273 207, 109, 63, 40, 18;
%e A363273 ...
%o A363273 (PARI)
%o A363273 T(n)={my(A=vector(n), R=vector(n)); A[1]=1; R[1]=[]; for(i=2, n, my(t=vector(i\2, j, if(2*j<i, A[j] * A[i-j], A[i/2] * (A[i/2] + 1)/2))); forstep(j=i\2-1, 1, -1, t[j]+=t[j+1]); A[i]=t[1]; R[i]=t); R}
%o A363273 { my(A=T(12)); for(n=2, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 01 2024
%Y A363273 First column k = 1 is A001190.
%Y A363273 Sums along upwards diagonals are A000671.
%Y A363273 Cf. A363272.
%K A363273 nonn,tabf
%O A363273 2,3
%A A363273 _Harry Richman_, May 24 2023
%E A363273 Terms a(27) and beyond from _Andrew Howroyd_, Jan 01 2024