OFFSET
0,2
COMMENTS
Row 1 of A364513.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k = 0..n} (-1)^k * binomial(n-1, n-k)^2 * binomial(n+1, k).
a(2*n) = 0 for n >= 1; a(2*n+1) = (-1)^(n+1) * 2/(2*n + 1) * (3*n + 1)!/n!^3.
a(2*n+1) ~ (-1)^(n+1) * 3^(3*n) * 3*sqrt(3)/(2*Pi*n).
P-recursive: a(0) = 1, a(1) = -2 and for n >= 2, a(n) = -(3*n - 1)*(3*n - 5)*(3*n - 6)/(n*(n - 1)^2) * a(n-2).
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.
a(n) = Sum_{k = 0..n} (-1)^k * binomial(n+k-1, k) * binomial(n-1, k) * binomial(2*n-k, n-k). - Peter Bala, Aug 13 2023
MAPLE
a := proc(n) option remember; if n = 0 then 1 elif n = 1 then -2 else -(3*n - 1)*(3*n - 5)*(3*n - 6)/(n*(n - 1)^2) * a(n-2) end if; end:
seq(a(n), n = 0..15);
MATHEMATICA
A364514[n_]:=A364514[n]=Which[n==0, 1, n==1, -2, True, -(3n-1)(3n-5)(3n-6)/(n(n-1)^2)A364514[n-2]]; Array[A364514, 40, 0] (* Paolo Xausa, Oct 05 2023 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Aug 02 2023
STATUS
approved