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

%I A361717 #37 Aug 03 2025 16:06:03
%S A361717 0,1,4,27,216,1875,17088,160867,1549936,15195843,151017780,1517232189,
%T A361717 15379549056,157058738343,1614039427224,16676755365555,
%U A361717 173118505001952,1804500885273123,18877476988765404,198120856336103017,2085303730716475960
%N A361717 a(n) = Sum_{k = 0..n-1} binomial(n-1,k)^2*binomial(n+k,k).
%C A361717 Compare with the Apery numbers A005258(n) = Sum_{k = 0..n} binomial(n,k)^2* binomial(n+k,k).
%C A361717 Conjecture 1: the supercongruence a(p) == 0 (mod p^4) holds for all primes p >= 5 (checked up to p = 199).
%C A361717 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).
%H A361717 Winston de Greef, <a href="/A361717/b361717.txt">Table of n, a(n) for n = 0..956</a>
%F A361717 a(n) = hypergeom([1 + n, 1 - n, 1 - n], [1, 1], 1) for n >= 1.
%F A361717 P-recursive:
%F A361717 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) = 0 and a(1) = 1.
%F A361717 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
%F A361717 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
%F A361717 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
%F A361717 From _Peter Bala_, Jul 31 2025: (Start)
%F A361717 a(n) = n * Sum_{k = 0..n} 1/(k+1) * binomial(n-1, k)^2 * binomial(n+k-1,k).
%F A361717 a(n) = n * hypergeom([n, 1 - n, 1 - n], [1, 2], 1). (End)
%e A361717 a(5) = 3*(5^4); a(7) = (7^4)*67; a(11) = 3*(11^4)*34543; a(13) = (3^3)*(13^4)*203669.
%p A361717 seq( add(binomial(n-1,k)^2*binomial(n+k,k), k = 0..n), n = 0..20);
%t A361717 A361717[n_]:=Sum[Binomial[n-1,k]^2Binomial[n+k,k],{k,0,n-1}];Array[A361717,30,0] (* _Paolo Xausa_, Oct 06 2023 *)
%o A361717 (PARI) a(n) = sum(k=0, n-1, binomial(n-1,k)^2*binomial(n+k,k)) \\ _Winston de Greef_, Mar 27 2023
%Y A361717 Cf. A005258, A006480, A060542, A361712, A361713, A361714, A361715.
%K A361717 nonn,easy
%O A361717 0,3
%A A361717 _Peter Bala_, Mar 26 2023