A361717 a(n) = Sum_{k = 0..n-1} binomial(n-1,k)^2*binomial(n+k,k).

0, 1, 4, 27, 216, 1875, 17088, 160867, 1549936, 15195843, 151017780, 1517232189, 15379549056, 157058738343, 1614039427224, 16676755365555, 173118505001952, 1804500885273123, 18877476988765404, 198120856336103017, 2085303730716475960

Offset

0,3

Comments

Compare with the Apery numbers A005258(n) = Sum_{k = 0..n} binomial(n,k)^2* binomial(n+k,k).

Conjecture 1: the supercongruence a(p) == 0 (mod p^4) holds for all primes p >= 5 (checked up to p = 199).

Conjecture 2: the supercongruence a(p-1) == 1 - 2*p - p^2 (mod p^3) holds for all primes except p = 3 (checked up to p = 199).

Links

Winston de Greef, Table of n, a(n) for n = 0..956

Formula

a(n) = hypergeom([1 + n, 1 - n, 1 - n], [1, 1], 1) for n >= 1.

P-recursive:

n*(n-1)*(5*n-7)*a(n) = (55*n^3-187*n^2+190*n-48)*a(n-1) + (n-1)*(n-3)*(5*n-2)* a(n-2) with a(0) = a(1) = 1.

a(n) ~ phi^(5*n - 3/2) / (2*5^(1/4)*Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 27 2023

a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1) * binomial(n-1, k) * binomial(n+k-1, k) * binomial(n+k, k+1) = (-1)^(n+1) * n * hypergeom([n, n + 1, 1 - n], [1, 2], 1). - Peter Bala, Sep 08 2023

a(n) = Sum_{k = 0..n-1} (-1)^k * binomial(n-2, k) * binomial(2*n-2-k, n-1-k)^2. - Peter Bala, Oct 09 2024

Example

a(5) = 3*(5^4); a(7) = (7^4)*67; a(11) = 3*(11^4)*34543; a(13) = (3^3)*(13^4)*203669.

Maple

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

Mathematica

A361717[n_]:=Sum[Binomial[n-1, k]^2Binomial[n+k, k], {k, 0, n-1}]; Array[A361717, 30, 0] (* Paolo Xausa, Oct 06 2023 *)

Prog

(PARI) a(n) = sum(k=0, n-1, binomial(n-1, k)^2*binomial(n+k, k)) \\ Winston de Greef, Mar 27 2023

Crossrefs

Cf. A005258, A006480, A060542, A361712, A361713, A361714, A361715.

Sequence in context: A059391 A371786 A190738 * A275607 A319518 A304045

Adjacent sequences: A361714 A361715 A361716 * A361718 A361719 A361720

Keyword

nonn,easy

Author

Peter Bala, Mar 26 2023

Status

approved