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A361712 a(n) = Sum_{k = 0..n-1} binomial(n,k)^2*binomial(n+k,k)*binomial(n+k-1,k).

%I A361712 #35 Mar 26 2025 08:31:50
%S A361712 0,1,25,649,16921,448751,12160177,336745053,9513822745,273585035755,
%T A361712 7988828082775,236367018090017,7072779699975601,213701611408357567,
%U A361712 6511338458568750853,199850727914988936149,6173376842290368719385,191776434791965521115235,5987554996434696230487955
%N A361712 a(n) = Sum_{k = 0..n-1} binomial(n,k)^2*binomial(n+k,k)*binomial(n+k-1,k).
%C A361712 Conjecture 1: the supercongruence a(p) == a(1) (mod p^5) holds for all primes p >= 7 (checked up to p = 199).
%C A361712 Conjecture 2: for r >= 2, the supercongruence a(p^r) == a(p^(r-1)) (mod p^(3*r+3)) holds for all primes p >= 5.
%C A361712 Compare with the Apéry numbers A005259(n) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(n+k,k)^2, which satisfy the weaker supercongruences A005259(p^r) == A005259(p^(r-1)) (mod p^(3*r)) for all primes p >= 5.
%H A361712 Paolo Xausa, <a href="/A361712/b361712.txt">Table of n, a(n) for n = 0..650</a>
%H A361712 Peter Bala, <a href="/A361712/a361712.pdf">Recurrence equation for A361712</a>
%F A361712 a(n) = (1/12)*(7*A005259(n) + A005259(n-1)) - (1/2)*binomial(2*n,n)^2.
%F A361712 a(n) ~ 2^(1/4)*(1 + sqrt(2))^(4*n+1)/(4*Pi^(3/2)*n^(3/2)).
%F A361712 a(n) = hypergeom([-n, -n, n, n + 1], [1, 1, 1], 1) - binomial(2*n, n)*binomial(2*n - 1, n) = A361878(n) - A361877(n). - _Peter Luschny_, Mar 27 2023
%e A361712 a(7) - a(1) = (2^2)*(7^5)*5009 == 0 (mod 7^5)
%e A361712 a(11) - a(1) = (2^5)*(11^5)*45864163 == 0 (mod 11^5)
%e A361712 a(7^2) - a(7) = (2*3)*(7^9)*377052719*240136524699189343838527* 17965610580703155723668147409587 == 0 (mod 7^9)
%p A361712 seq(add(binomial(n,k)^2*binomial(n+k,k)*binomial(n+k-1,k), k = 0..n-1), n = 0..25);
%p A361712 # Alternative:
%p A361712 A361712 := n -> hypergeom([-n, -n, n, n + 1], [1, 1, 1], 1) - binomial(2*n, n)*binomial(2*n-1, n): seq(simplify(A361712(n)), n = 0..18); # _Peter Luschny_, Mar 27 2023
%t A361712 A361712[n_] := HypergeometricPFQ[{-n, -n, n, n+1}, {1, 1, 1}, 1] - Binomial[2*n, n]*Binomial[2*n-1, n]; Array[A361712, 20, 0] (* _Paolo Xausa_, Jul 10 2024 *)
%Y A361712 Cf. A005259, A212334, A361713, A361714, A361715, A361717.
%Y A361712 Cf. A361877, A361878.
%K A361712 nonn,easy
%O A361712 0,3
%A A361712 _Peter Bala_, Mar 21 2023