A365029 a(n) = Sum_{k = 0..n} binomial(n+k-1, k)^2 * binomial(2*k-1, n).

1, 0, 28, 1035, 44876, 2104500, 104056597, 5342503859, 282118965580, 15225746918238, 836111285393528, 46569126655126867, 2624469492691484309, 149381829558924820091, 8575171411278263451149, 495882491862054255448035, 28860386333798348100899148, 1689200944709783371200111774

Offset

0,3

Comments

Conjecture: for all primes p >= 5 the following pair of supercongruences hold:

1) a(p - 1) == a(0) (mod p^3),

2) a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) for all positive integers n and r.

More generally, for positive integers A and B with A >= 2, the same supercongruences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(n+k-1, k)^A * binomial(2*k-1, n)^B.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) ~ 2^(6*n+1) / (15*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Aug 28 2023

Maple

seq( add( binomial(n+k-1, k)^2 * binomial(2*k-1, n), k = 0..n), n = 0..20);

Mathematica

Table[Sum[Binomial[n+k-1, k]^2 * Binomial[2*k-1, n], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 28 2023 *)

Crossrefs

Sequence in context: A228689 A218480 A162006 * A370358 A025753 A160312

Adjacent sequences: A365026 A365027 A365028 * A365030 A365031 A365032

Keyword

nonn,easy

Author

Peter Bala, Aug 27 2023

Status

approved