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A357558
a(n) = Sum_{k = 0..n} (-1)^(n+k)*k*binomial(n,k)*binomial(n+k,k)^2.
3
0, 4, 54, 648, 7500, 85440, 965202, 10849552, 121566744, 1359160020, 15172321890, 169175039616, 1884704860116, 20982512553912, 233474575117770, 2596777575029280, 28872014164369968, 320917108809011868, 3566175414049854306, 39620770883613043240, 440115513924937822020
OFFSET
0,2
FORMULA
Conjecture: a(p-1) == 0 (mod (p - 1)*p^4) for all primes p >= 5 (checked up to p = 499).
Note: Let B(n) = A005258(n). It is known that B(n) = Sum_{k = 0..n} (-1)^(n+k)* binomial(n,k)*binomial(n+k,k)^2 and the supercongruences B(p-1) == 1 (mod p^3) hold for all primes p >= 5 (see, for example, Straub, Example 3.4).
Recurrence: a(0) = 0, a(1) = 4 and for n >= 2, (5*n - 2)*(n^2 - 1)*a(n) = (55*n^3 - 22*n^2 - 19*n + 10)*a(n-1) + n*(5*n + 3)*(n-1)*a(n-2).
a(n) ~ phi^(5*n + 7/2) / (2*Pi*5^(1/4)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 05 2022
EXAMPLE
Example of a supercongruence:
p = 17: a(17 - 1) = 28872014164369968 = (2^4)*3*(17^4)*107*251*268153 == 0 (mod 16*7^4)
MAPLE
seq( add( (-1)^(n+k)*k*binomial(n, k)*binomial(n+k, k)^2, k = 0..n ), n = 0..20 );
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Oct 03 2022
STATUS
approved