OFFSET
0,2
COMMENTS
Related to planar maps? - see A000184. - N. J. A. Sloane, Mar 11 2007
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = 4^n * Sum_{k=0..n} A000531(k+1)/4^k.
a(n) = (1/3)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n) - (n+2)*2^(2*n+1).
a(n) = 4*a(n-1) + A000531(n+1).
G.f. c(x)/(1-4*x)^(5/2) = (2-c(x))/(1-4*x)^3, where c(x) = g.f. for Catalan numbers; also convolution of Catalan numbers with A002802.
G.f.: (4*x-1+sqrt(1-4*x))/(2*x*(1-4*x)^3). - Vincenzo Librandi, Mar 14 2014
From G. C. Greubel, Jul 18 2024:
a(n) = (1/24)*(n+2)*((n+3)*(n+4)*Catalan(n+3) - 3*4^(n+2)).
a(n) = (1/2)*A000184(n+2). (End)
MATHEMATICA
a[n_] := (2*n+1)*(2*n+3)*(2*n+5)*CatalanNumber[n]/3 - (n+2)*2^(2*n+1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 12 2014 *)
CoefficientList[Series[(4 x - 1 + Sqrt[1 - 4 x])/(2 x (1 - 4 x)^3), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 14 2014 *)
PROG
(Magma) [(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n)/3 - (n+2)*2^(2*n+1): n in [0..30]]; // Vincenzo Librandi, Mar 14 2014
(SageMath)
[(n+2)*((n+3)*(n+4)*catalan_number(n+3) - 3*4^(n+2))//24 for n in range(31)] # G. C. Greubel, Jul 18 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vincenzo Librandi, Mar 14 2014
STATUS
approved