A138156 Sum of the path lengths of all binary trees with n edges.

0, 2, 14, 74, 352, 1588, 6946, 29786, 126008, 527900, 2195580, 9080772, 37392864, 153434536, 627778954, 2562441466, 10438340104, 42449348236, 172376641924, 699100282156, 2832205421824, 11462854280536, 46354571222164

Offset: 0

Author

Emeric Deutsch, Mar 20 2008

Keywords

• nonn

Comments

a(n) = 2*A006419(n).

If (2*n+3) prime, then A138156(n) mod (2*n+3) == 0. - Alzhekeyev Ascar M, Jul 19 2011

Examples

			a(1) = 2 because the trees with one edge are (i) root with a left child and (ii) root with a right child, each having path length 1.
		

References

• D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1997, Vol. 1, p. 405 (exercise 5) and p. 595 (solution).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1659
• Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.

Crossrefs

Cf. A095830, A138157, A006419.

Programs

Maple

a:= n-> 4^(n+1)-(3*n+4)*binomial(2*n+2, n+1)/(n+2): seq(a(n), n=0..22);

Mathematica

Table[4^(n+1)-(3n+4) Binomial[2n+2,n+1]/(n+2),{n,0,30}] (* Harvey P. Dale, Dec 14 2014 *)

Formula

a(n) = 4^(n+1) - (3*n+4) * C(2*n+2,n+1)/(n+2).

G.f.: 1/(z*(1-4*z)) - ((1-z)/sqrt(1-4*z)-1)/z^2.

D-finite with recurrence (n+2)*a(n) +(-9*n-10)*a(n-1) +2*(12*n+1)*a(n-2) +8*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jul 26 2022