OFFSET
0,2
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = 4 * A047781(n).
a(n) = A102413(2*n,n).
a(n) = 2*Hyper2F1([-n, n], [1], -1) for n>0. - Peter Luschny, Aug 02 2014
D-finite g.f. = (1+x)/sqrt(1-6*x+x^2), pairwise sums of A001850. - R. J. Mathar, Jan 15 2020
From Peter Bala, Apr 16 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*(2^k)*binomial(2*k, k)*binomial(n+k-1, n-k).
a(n) = (-1)^(n+1) * 4*n * hypergeom([n+1, -n+1], [2], 2).
n*(2*n - 3)*a(n) = 4*(3*n^2 - 6*n + 2)*a(n-1) - (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1 and a(1) = 4.
O.g.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)*x^n/(1 + x)^(2*n) = 1 + 4*x + 16*x^2 + 76*x^3 + 384*x^4 + .... (End)
From Peter Bala, Sep 18 2024: (Start)
a(n) = [x^n] 1/S(-x)^(2*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318. Cf. A333481.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. (End)
MATHEMATICA
a[0] = 1; a[n_] := 4 Hypergeometric2F1[1 - n, n + 1, 1, -1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 28 2019 *)
PROG
(Haskell)
a241023 n = a102413 (2 * n) n
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Apr 15 2014
STATUS
approved