OFFSET
1,2
COMMENTS
On Jun 14 2010, Zhi-Wei Sun conjectured that a(n) = (1/n^2) * Sum_{k = 0..n-1} (2*k+1)*(D_k)^2 is always an integer and that p^2*a(p) = p^2 - 4*p^3*q_p(2) - 2*p^4*q_p(2)^2 (mod p^5) for any prime p > 3, where q_p(2) denotes the Fermat quotient (2^(p-1) - 1)/p (see Sun, Remark 4.3, p. 26, 2014). He also conjectured that Sum_{k = 0..n-1} (2*k+1)*(-1)^k*(D_k)^2 == 0 (mod n*D_n/(3,D_n)) for all n = 1,2,3,....
The fact that a(n) is an integer follows directly from the formulas for a(n) in the formula section below. - Mark van Hoeij, Nov 13 2022
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2011.
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57 (2014), no. 7, 1375-1400; arXiv:1008.3887 [math.NT], 2010-2014.
FORMULA
a(n) ~ (1 + sqrt(2))^(4*n) / (16*Pi*n^2). - Vaclav Kotesovec, Jan 24 2019
G.f.: Integral(hypergeom([1/2, 1/2], [2], -32*x/(1 - 34*x + x^2))/((1 - x)*(1 - 34*x + x^2)^(1/2))). - Mark van Hoeij, Nov 10 2022
a(n) = (3*f(n)*f(n-1) - g(n))/4, where g(n) = hypergeom([n, -n, 1/2], [1, 1], -8) and f(n) = hypergeom([-n, -n], [1], 2). This formula also gives an integer value for n = 0. - Peter Luschny, Nov 13 2022
EXAMPLE
For n = 3 we have a(3) = (D_0^2 + 3*D_1^2 + 5*D_2^2)/3^2 = (1 + 3*3^2 + 5*13^2)/3^2 = 97.
MAPLE
A001850 := n -> LegendreP(n, 3); seq((6*A001850(n)*A001850(n-1)-A001850(n)^2-A001850(n-1)^2)/8, n=1..20); # Mark van Hoeij, Nov 12 2022
# Alternative:
g := n -> hypergeom([n, -n, 1/2], [1, 1], -8): # A358388
f := n -> hypergeom([-n, -n], [1], 2): # A001850
a := n -> (3*f(n)*f(n-1) - g(n)) / 4:
seq(simplify(a(n)), n = 1..17); # Peter Luschny, Nov 13 2022
MATHEMATICA
DD[n_]:=Sum[Binomial[n+k, 2k]Binomial[2k, k], {k, 0, n}]; SS[n_]:= Sum[(2k+1)*DD[k]^2, {k, 0, n-1}]/n^2; Table[SS[n], {n, 1, 25}]
Table[Sum[(2k+1)*JacobiP[k, 0, 0, 3]^2, {k, 0, n-1}]/n^2, {n, 1, 30}] (* G. C. Greubel, Jan 23 2019 *)
PROG
(Python) # prepends a(0) = 0
def A178808List(size: int) -> list[int]:
A358387 = A358387gen()
A358388 = A358388gen()
print(A178808List(18)) # Peter Luschny, Nov 15 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 16 2010
STATUS
approved