OFFSET
0,4
COMMENTS
Hankel transform is (-2)^C(n+1,2). - Paul Barry, Apr 28 2009
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (-(1+x)+sqrt(1+2*x+9*x^2))/(4*x^2). - Corrected by Seiichi Manyama, May 12 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*C(k)*(-1)^(n-k)*2^k, where C(n) is A000108(n). - Paul Barry, May 16 2005
G.f.: 1/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+ ... (continued fraction). - Paul Barry, Apr 28 2009
a(n) = Sum_{i=0..n} (2^(i)*(-1)^(n-i)*binomial(n+1,i)^2*(n-i+1)/(i+1))/(n+1). - Vladimir Kruchinin, Oct 12 2011
Conjecture: (n+2)*a(n) +(2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
a(n) = (-1)^n*hypergeom([-n/2, (1-n)/2], [2], -8). - Peter Luschny, May 28 2014
R. J. Mathar's conjecture confirmed by Maple using this hypergeom form. - Robert Israel, Sep 22 2014
a(n) = Sum_{k = 0..n} (-2)^k * (1/(n+1))*binomial(n+1, k)*binomial(n+1, k+1) = Sum_{k = 0..n} (-2)^k * N(n+1,k+1), where N(n,k) = A001263(n,k) are the Narayana numbers. - Peter Bala, Sep 01 2023
MAPLE
a := n -> hypergeom([-n, -n-1], [2], -2);
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 22 2014
# Using function CompInv from A357588.
CompInv(25, n -> (2^n - (-1)^n)/3 ); # Peter Luschny, Oct 07 2022
MATHEMATICA
a[n_] := Hypergeometric2F1[-n - 1, -n - 1, 2, -2] + (n + 1)*Hypergeometric2F1[-n, -n, 3, -2]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 03 2016, after Vladimir Kruchinin *)
PROG
(Maxima)
a(n):=sum(2^(i)*(-1)^(n-i)*binomial(n+1, i)^2*(n-i+1)/(i+1), i, 0, n)/(n+1); (* Vladimir Kruchinin, Oct 12 2011 *)
(Sage) # Algorithm of L. Seidel (1877)
def A091593_list(n) :
D = [0]*(n+2); D[1] = 1
R = []; b = false; h = 1
for i in range(2*n) :
if b :
for k in range(1, h, 1) : D[k] += -2*D[k+1]
R.append(D[1])
else :
for k in range(h, 0, -1) : D[k] += D[k-1]
h += 1
b = not b
return R
A091593_list(30) # Peter Luschny, Oct 19 2012
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Jan 23 2004
STATUS
approved