OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 18.
FORMULA
G.f.: (2*x-1+sqrt((1-2*x)*(1-6*x+4*x^2)))/(2*x*(x-1)).
G.f.: W(0), where W(k) = 1 + (4*k+1)*x*(1-x)/( (k+1)*(1-2*x) - 2*x*(1-x)*(1-2*x)*(k+1)*(4*k+3)/(2*x*(1-x)*(4*k+3) + (2*k+3)*(1-2*x)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 12*(2*n-3)*a(n-2) + 12*(2*n-5)*a(n-3) - 4*(2*n-7)*a(n-4). - Vaclav Kotesovec, Jun 19 2014
a(n) ~ sqrt(10-2*sqrt(5)) * (3+sqrt(5))^n / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 19 2014. Equivalently, a(n) ~ 5^(1/4) * 2^n * phi^(2*n - 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
a(n) = Sum_{k=0..n} ((C(k)*Sum_{i=0..n-k} (2^i*binomial(k,n-k-i)*binomial(k+i-1,i)*(-1)^(n-k-i))))), where C(k) is the k-th Catalan number. - Vladimir Kruchinin, Mar 04 2016
MAPLE
f:= gfun:-rectoproc({(4+8*n)*a(n)+(-36-24*n)*a(1+n)+(60+24*n)*a(n+2)+(-33-9*n)*a(n+3)+(5+n)*a(n+4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 11}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 04 2016
MATHEMATICA
CoefficientList[Series[(2*x-1+Sqrt[(1-2*x)*(1-6*x+4*x^2)])/(2*x*(x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 19 2014 *)
Table[Sum[CatalanNumber[k] Sum[2^i Binomial[k, n - k - i] Binomial[k + i - 1, i] (-1)^(n - k - i), {i, 0, n - k}], {k, 0, n}], {n, 0, 24}] (* Michael De Vlieger, Mar 04 2016 *)
PROG
(PARI) a(n)=polcoeff((2*x-1 +sqrt((1-2*x)*(1-6*x+4*x^2)+x^2*O(x^n))) /(2*x^2-2*x), n);
(PARI) x='x+O('x^100); Vec((2*x-1+sqrt((1-2*x)*(1-6*x+4*x^2)))/(2*x*(x-1))) \\ Altug Alkan, Mar 05 2016
(Maxima)
a(n):=sum((binomial(2*k, k)*sum(2^i*binomial(k, n-k-i)*binomial(k+i-1, i)*(-1)^(n-k-i), i, 0, n-k))/(k+1), k, 0, n); /* Vladimir Kruchinin, Mar 04 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 24 2001
STATUS
approved