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

0,4

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

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A039991 A273821 A363519 * A133838 A182138 A258123

Adjacent sequences: A108640 A108641 A108642 * A108644 A108645 A108646

Keyword

nonn

Author

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

Extensions

More terms from Vladeta Jovovic, Jul 08 2005

Status

approved