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
KEYWORD
nonn
AUTHOR
Louis Shapiro, Aug 21 2013
STATUS
approved