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A108643
Number of binary rooted trees with n nodes and internal path length n.
6
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jul 08 2005
STATUS
approved