OFFSET
0,2
COMMENTS
The Hankel transform of this sequence is 20^C(n+1,2).
a(n) is the number of small Schröder n-paths with 4 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=U4=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 05 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv preprint, arXiv:1912.00555 [math.CO], 2019.
FORMULA
a(n) = Sum_{k=0..n} A086810(n,k)*4^k.
From Paul Barry, Sep 08 2009: (Start)
a(n) = Sum_{k=0..n} C(n+k,2*k)*A000108(k)*(-1)^(n-k)*5^k;
a(n) = Sum_{k=0..n} C(n+k,2*k)*A000108(k)*(4^(k+1)+(-1)^k)/5. (End)
Recurrence: (n+1)*a(n) = 9*(2*n-1)*a(n-1) - (n-2)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(40+18*sqrt(5))*(9+4*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012. Equivalently, a(n) ~ phi^(6*n + 3) / (sqrt(2) * 5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -4). - Peter Luschny, Jan 08 2018
MATHEMATICA
Table[Sum[Binomial[n+k, 2*k]*Binomial[2*k, k]/(k+1)*(-1)^(n-k)*5^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := Sum[(-1)^k Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4], {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Peter Luschny, Jan 08 2018 *)
PROG
(PARI) Vec(serreverse(x*(1-5*x)/(1-x) + O(x^30))) \\ Michel Marcus, Jan 08 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, Oct 29 2007
EXTENSIONS
Extra terms added by Paul Barry, Sep 08 2009
STATUS
approved