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A228404 The number of complete binary trees with bicolored twigs. A twig is a vertex with one child on the boundary and the other child having no descendants. 2
1, 2, 8, 24, 76, 249, 836, 2860, 9932, 34918, 124032, 444448, 1604664, 5831765, 21316860, 78319140, 289064460, 1071275370, 3984871440, 14872552560, 55678270440, 209027020410, 786750047304, 2968257334104, 11223268563896, 42522737574604, 161415556062656, 613813414982656 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

FORMULA

G.f.: 1 - x + 2*x*C^2 + x*C^4 where C is the g.f. for the Catalan numbers A000108.

Conjecture: -5*(n+3)*(n-2)*a(n) +5*(-n^2-n+18)*a(n-1) +5*(-n^2-n+48)*a(n-2) +(-5*n^2+20029*n+720)*a(n-3) +(-5*n^2-104153*n+186654)*

a(n-4) +(-5*n^2 +130153*n -508806)*a(n-5) +13650*(2*n-11)*(n-7)*a(n-6) = 0. - R. J. Mathar, Aug 08 2015

From G. C. Greubel, May 03 2021: (Start)

a(n) = C(n+2) - 2*C(n+1) + 2*C(n) with a(0) = 1, a(1) = 2, and C(n) = A000108(n).

E.g.f.: (-x^2*(1+x) + 2*exp(2*x) *( x*(1+x)*BesselI(0, 2*x) - (1+x^2)*BesselI(1, 2*x))/x^2. (End)

EXAMPLE

For n = 2 there are two complete binary trees. Both consist of two twigs so can be colored 4 ways each.

MATHEMATICA

Table[If[n<2, n+1, CatalanNumber[n+2] -2*CatalanNumber[n+1] +2*CatalanNumber[n]], {n, 0, 30}] (* G. C. Greubel, May 03 2021 *)

PROG

(PARI)

x = 'x + O('x^66);

C = serreverse( x/( 1/(1-x) ) ) / x; \\ Catalan A000108

gf = 1 - x + 2*x*C^2 + x*C^4;

Vec(gf) \\ Joerg Arndt, Aug 22 2013

(Magma) [1, 2] cat [Catalan(n+2) -2*Catalan(n+1) +2*Catalan(n): n in [2..30]]; // G. C. Greubel, May 03 2021

(Sage) [1, 2]+[catalan_number(n+2) -2*catalan_number(n+1) +2*catalan_number(n) for n in (2..30)] # G. C. Greubel, May 03 2021

CROSSREFS

Without the bicoloring A228403 is the result.

Cf. A000108.

Sequence in context: A130495 A026070 A093833 * A006952 A327550 A034741

Adjacent sequences:  A228401 A228402 A228403 * A228405 A228406 A228407

KEYWORD

nonn

AUTHOR

Louis Shapiro, Aug 21 2013

STATUS

approved

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Last modified June 23 07:38 EDT 2021. Contains 345395 sequences. (Running on oeis4.)