A108643 Number of binary rooted trees with n nodes and internal path length n.

1, 1, 0, 2, 0, 1, 4, 0, 4, 2, 8, 6, 8, 8, 8, 40, 4, 29, 40, 52, 56, 64, 116, 112, 200, 86, 296, 366, 360, 432, 652, 800, 840, 1470, 1116, 2048, 2356, 3052, 3524, 4220, 5648, 6964, 9660, 8688, 14128, 17024, 19432, 23972, 32784, 37873, 44912, 59672, 67560, 93684

Offset: 0

Author

N. J. A. Sloane and Nadia Heninger, Jul 08 2005

Keywords

• nonn

Comments

Self-convolution equals A095830 (number of binary trees of path length n). - Paul D. Hanna, Aug 20 2007

References

• Knuth Vol. 1 Sec. 2.3.4.5, Problem 5.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Programs

Maple

A:= proc(n,k) option remember; if n=0 then 1 else convert(series(1+ x^k*A(n-1, k+1)^2, x,n+1), polynom) fi end: a:= n-> coeff(A(n,1), x,n): seq(a(n), n=0..60);  # Alois P. Heinz, Aug 22 2008

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, 1+x^k*A[n-1, k+1]^2 + O[x]^(n+1) // Normal]; a[n_] := SeriesCoefficient[A[n, 1], {x, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 14 2017, after Alois P. Heinz *)

PARI

{a(n)=local(A=1+x*O(x^n)); for(j=0,n-1,A=1+x^(n-j)*A^2);polcoeff(A,n)} \\ Paul D. Hanna, Aug 20 2007

Formula

G.f. = B(w, w) where B(w, z) is defined in A095830.

G.f.: A(x) = 1 + x*(A_2)^2; A_2 = 1 + x^2*(A_3)^2; A_3 = 1 + x^3*(A_4)^2; ... A_n = 1 + x^n*(A_{n+1})^2 for n>=1 with A_1 = A(x). - Paul D. Hanna, Aug 20 2007

Extensions

More terms from Vladeta Jovovic, Jul 08 2005