OFFSET
0,3
COMMENTS
This is row n=8 in the array A(n,k) = (rf(k+n-2,k-1)-(k-1)*(k-2)*rf(k+n-2, k-3))/ (k-1)! if n>=3 and A(n,0)=0, A(n,1)=1, A(n,2)=n; rf(n,k) denotes the rising factorial. See the cross-references for other values of n and the table in A264357.
LINKS
Robert Israel, Table of n, a(n) for n = 0..1630
FORMULA
G.f.: I*(14*x^2+I*sqrt(4*x-1)*(4*x^2-7*x+2)-11*x+2*(1-x^3))/(2*x^4*sqrt(4*x-1)).
a(n) = (4^(n+1)*n*(n+1)*(3*n+2)*Gamma(n+3/2))/(sqrt(Pi)*Gamma(n+5)).
a(n) = (rf(n+6, n-1)-(n-1)*(n-2)*rf(n+6, n-3))/(n-1)! for n>=3, rf(n,k) the rising factorial.
a(n) = a(n-1)*((2*(n+1))*(3*n+2)*(1+2*n)/((n-1)*(3*n-1)*(4+n))) for n>=2.
a(n) ~ 4^n*(12-(191/2)/n+(17595/32)/n^2-(705005/256)/n^3+(104705937/8192)/ n^4-...)/sqrt(n*Pi).
a(n) = [x^n] x*(1 + x)/(1 - x)^(n+5). - Ilya Gutkovskiy, Oct 09 2017
MAPLE
MATHEMATICA
Table[SeriesCoefficient[I (14 x^2 + I Sqrt[4 x - 1] (4 x^2 - 7 x + 2) - 11 x + 2 (1 - x^3))/(2 x^4 Sqrt[4 x - 1]), {x, 0, n}], {n, 0, 25}]
(* or *)
Table[(4^(n + 1) n (n + 1) (3 n + 2) Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 5]), {n, 0, 25}] (* or *)
Table[CatalanNumber(n+1) n (3 n^2 + 5 n + 2)/((4 + n) (3 + n)), {n, 0, 25}] (* Michael De Vlieger, Dec 15 2015 *)
PROG
(Sage)
a = lambda n: catalan_number(n+1)*n*(3*n^2+5*n+2)/((4+n)*(3+n))
[a(n) for n in range(26)]
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Dec 15 2015
STATUS
approved