OFFSET
1,1
LINKS
R. H. Hardin, Table of n, a(n) for n=1..35
FORMULA
From Peter Bala, Nov 05 2024: (Start)
The following are conjectural:
For n >= 1, a(n) = Sum_{k = 0..2*n} (-1)^(n+k) * (k/n)^2 * binomial(2*n, k)^4. Cf. the identity Sum_{k = 0..2*n} (-1)^(n+k) * (k/n) * binomial(2*n, k)^2 = binomial(2*n, n) = A000984(n) for n >= 1.
For n >= 1, a(n) = 2 * binomial(2*n, n) * Sum_{k = 0..n} (k/n) * binomial(2*n, n-k)^2 * binomial(2*n+k, k).
P-recursive: n^3*(2*n-1)*(n-1)*(24*n^3-105*n^2+152*n-73)*a(n) = 2*(n-1)*(3264*n^7-20808*n^6+53900*n^5-73159*n^4+55963*n^3-24107*n^2+5436*n-504)*a(n-1) - 4*(2*n-1)*(24*n^3-33*n^2+14*n-2)*(2*n-3)^2*(n-2)^2*a(n-2) with a(1) = 12 and a(2) = 660.
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.(End)
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
R. H. Hardin, May 06 2010
STATUS
approved