OFFSET
1,3
COMMENTS
Conjecture: (i) All the terms are odd integers. For any prime p, if p == 3 (mod 4) then a(p) == -5 (mod p^2), otherwise a(p) == -1 (mod p).
(ii) For n = 0,1,2,... let D_n(x) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k and R_n(x) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k/(2k-1). For any positive integer n, all the coefficients of the polynomial (1/n)*Sum_{k=0..n-1} D_k(x)*R_k(x) are integral and the polynomial is irreducible over the field of rational numbers.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Two new kinds of numbers and related divisibility results, preprint, arXiv:1408.5381 [math.NT], 2014.
Zhi-Wei Sun, Arithmetic properties of Delannoy numbers and Schröder numbers, preprint, arXiv:1602.00574 [math.CO], 2016.
EXAMPLE
MATHEMATICA
d[n_]:=d[n]=Sum[Binomial[n, k]Binomial[n+k, k], {k, 0, n}]
R[n_]:=R[n]=Sum[Binomial[n, k]Binomial[n+k, k]/(2k-1), {k, 0, n}]
a[n_]:=a[n]=Sum[d[k]*R[k], {k, 0, n-1}]/n
Do[Print[n, " ", a[n]], {n, 1, 20}]
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Jan 26 2016
STATUS
approved