OFFSET
2,1
COMMENTS
Conjecture 1: a(n) is a positive integer for each n > 1. Moreover, a(n) is odd if and only if n = 2^k + 1 for some nonnegative integer k.
Conjecture 2: The infinite series Sum_{k>=0} (4*k+1)/(-160)^k * C(2k,k) * Sum_{j=0..k} C(k,j) * C(k+2j,2j) * C(2j,j) * (-20)^(k-j) has the value sqrt(30)/(5*Pi)*(5+c^(1/3))/c^(1/6), where c = 145 + 30*sqrt(6).
Conjecture 3. Let p be an odd prime different from 5, and let S(p) denote the sum Sum_{k=0..p-1} C(2k,k)/(-160)^k * Sum_{j=0..k} C(k,j) * C(k+2j,2j) * C(2j,j) * (-20)^(k-j).
(i) If p == 1,3 (mod 8) and p = x^2 + 2*y^2 with x and y integers, then S(p) == (5/p)*(4x^2 - 2p) (mod p^2), where (5/p) is the Legendre symbol.
(ii) If p == 5,7 (mod 8), then S(p) == 0 (mod p^2).
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 2..100
Zhi-Wei Sun, Two curious series for 1/Pi, Question 369569 at MathOverflow, August 19-20, 2020.
Zhi-Wei Sun, New series for powers of Pi and related congruences, Electron. Res. Arch. 28(2020), no. 3, 1273-1342.
Zhi-Wei Sun, Some new series for 1/Pi motivated by congruences, arXiv:2009.04379 [math.NT], 2020.
EXAMPLE
a(2) = (160 - (4 + 1)*C(2,1)*(-20 + C(3,2)*C(2,1)))/(2*C(4,2)) = 300/12 = 25.
MATHEMATICA
a[n_]:=a[n]=Sum[(4k+1)(-1)^k*160^(n-1-k)*Binomial[2k, k]*Sum[Binomial[k, j]Binomial[k+2j, 2j]Binomial[2j, j](-20)^(k-j), {j, 0, k}], {k, 0, n-1}]/(n*Binomial[2n, n])
Table[a[n], {n, 2, 18}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 20 2020
STATUS
approved