A061639 Number of planar planted trees with n non-root nodes and every 2-valent node isolated.

0, 1, 1, 1, 4, 10, 28, 85, 262, 829, 2677, 8776, 29143, 97825, 331381, 1131409, 3889381, 13450744, 46764532, 163357807, 573064849, 2018027719, 7131064045, 25278463756, 89866690732, 320328538033, 1144591699069, 4099050204445, 14710315969696, 52893571881142

Offset

0,5

Comments

Isolated 2-valent node is a 2-valent node non-adjacent to any other 2-valent node.

Hankel transform of a(n+1) is A174108. Binomial transform of a(n+1) is A174107. - Paul Barry, Mar 07 2010

References

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.7.4).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).

Formula

a(n) = Sum_{m=0..n-1} Sum_{i=0..n-m-1} (-1)^i/(m+1)*binomial(2*m, m)*binomial(m+i, i)*binomial(m+i+1, n-m-i-1).

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

From Paul Barry, Mar 07 2010: (Start)

G.f.: (x(1+x)/(1+x+x^2))c(x(1+x)/(1+x+x^2)), c(x) the g.f. of A000108;

G.f.: (1+x+x^2-sqrt(1-2x-5x^2-6x^3-3x^4))/(2*(1+x+x^2)). (End)

Conjecture: n*a(n) +2*(2-n)*a(n-1) +(14-5*n)*a(n-2) +6*(3-n)*a(n-3) +3*(4-n)*a(n-4)=0. - R. J. Mathar, Nov 15 2011

a(n) ~ 7^(1/4) * 2^(n-7/2) * 3^(n+1/4) / (sqrt(Pi) * n^(3/2) * (sqrt(21)-3)^(n-1/2)). - Vaclav Kotesovec, Feb 12 2014

From Peter Bala, May 30 2017: (Start)

G.f. A(x) satisfies the differential equation (3*x^4 + 6*x^3 + 5*x^2 + 2*x - 1)*A(x)' - (4*x + 2)*A(x) + 2*x + 1 = 0 with A(0) = 0. Mathar's conjectural recurrence above follows from this.

1 - A(x) + A(x)^2 = 1/(1 + x + x^2). (End)

From Nikolaos Pantelidis, Apr 11 2022: (Start)

G.f.: f(x) = (sqrt(1+x+x^2)-sqrt(1-3*x-3*x^2))/(2*sqrt(1+x+x^2)).

Series reversion of g.f. f(x) is -f(-x). (End) [Corrected by Joerg Arndt, Jun 08 2022]

Mathematica

CoefficientList[Series[1/2*(1-Sqrt[1-4*(x+x^2)/(1+x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Crossrefs

Sequence in context: A103457 A083587 A228403 * A243600 A239577 A008995

Adjacent sequences: A061636 A061637 A061638 * A061640 A061641 A061642

Keyword

nonn,easy

Author

Vladeta Jovovic, Jun 13 2001

Status

approved