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.

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, 63, 40, 18, 451, 244, 146, 100, 54, 21, 983, 532, 325, 227, 135, 66, 2179, 1196, 745, 538, 342, 204, 66, 4850, 2671, 1688, 1237, 823, 529, 253, 10905, 6055, 3876, 2893, 1991, 1370, 782, 276

Offset

2,3

Links

Andrew Howroyd, Table of n, a(n) for n = 2..2501 (rows 1..100)

Formula

T(n,k) = Sum_{j >= k} A363272(n,j).

Sum_{k >= 1} T(n-k, k) = A000671(n-2).

Example

Table begins:
  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, 63, 40, 18;
...

Prog

(PARI)
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}
{ my(A=T(12)); for(n=2, #A, print(A[n])) } \\ Andrew Howroyd, Jan 01 2024

Crossrefs

First column k = 1 is A001190.

Sums along upwards diagonals are A000671.

Cf. A363272.

Sequence in context: A006241 A336105 A282601 * A034869 A205858 A340793

Adjacent sequences: A363270 A363271 A363272 * A363274 A363275 A363276

Keyword

nonn,tabf

Author

Harry Richman, May 24 2023

Extensions

Terms a(27) and beyond from Andrew Howroyd, Jan 01 2024

Status

approved