|
|
|
|
1, 253, 494341, 1403375905, 4684608730309, 17126002734202253, 66366682204430084569, 267832273159817887638881, 1113652383352571992799711941, 4737943697041408831021629805753, 20526206833382185439454748049996341
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
a(n) = A(3*n,2*n,3*n,2*n) in the notation of Straub, equation 8. It follows from Straub, Theorem 1.2, that the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and all positive integers n and k.
More generally, for positive integers r and s, the sequence {A143007(r*n, s*n) : n >= 0} satisfies the above supercongruences. For other cases, see A005259 (r = s = 1), A363864 (r = 2, s = 1) and A363865 (r = 3, s = 1).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{k = 0..2*n} binomial(3*n+k,2*k)*binomial(2*k,k)^2*binomial(2*n+k,2*k).
a(n) = Sum_{k = 0..2*n} binomial(3*n,k)^2*binomial(5*n-k,2*n-k)^2.
a(n) = hypergeom([3*n+1, -2*n, 3*n+1, -2*n], [1, 1, 1], 1)
a(n) = [x^(2*n)] 1/(1 - x)*( Legendre_P(3*n,(1 + x)/(1 - x)) )^2 = [x^(3*n)] 1/(1 - x)*( Legendre_P(2*n,(1 + x)/(1 - x)) )^2.
|
|
MAPLE
|
A143007 := proc(n, k); add(binomial(n+j, 2*j)*binomial(2*j, j)^2*binomial(k+j, 2*j), j = 0..n) end:
# alternative program
seq(simplify(hypergeom([3*n+1, -3*n, 2*n+1, -2*n], [1, 1, 1], 1)), n = 0..20);
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|