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.

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, 18, 207, 98, 46, 46, 33, 21, 451, 207, 98, 92, 69, 66, 983, 451, 207, 196, 138, 138, 66, 2179, 983, 451, 414, 294, 276, 253, 4850, 2179, 983, 902, 621, 588, 506, 276

Offset: 2

Author

Harry Richman, May 24 2023

Keywords

• nonn
• tabf

Examples

			Table begins:
 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, 18;
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..2501 (rows 2..100)

Crossrefs

Row sums are A001190.

First column k=1 is T(n,1) = A001190(n-1).

Cf. A000671, A363273.

Programs

PARI

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*jAndrew Howroyd, Jan 01 2024

Formula

T(n,k) = A001190(k) * A001190(n-k) if k < n/2; otherwise

T(2k,k) = A001190(k) * (A001190(k) + 1) / 2 = A000217(A001190(n)).

Sum_{k >= 1} T(n,k) = A001190(n).

Sum_{i >= k} T(n,i) = A363273(n,k).

Sum_{i <= n-1, i+j >= n} T(i,j) = A000671(n-2).

Extensions

Terms a(32) and beyond from Andrew Howroyd, Jan 01 2024