The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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