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A361878 a(n) = hypergeom([-n, -n, n, n + 1], [1, 1, 1], 1).

%I A361878 #14 Mar 29 2023 10:30:39
%S A361878 1,3,43,849,19371,480503,12587065,342634365,9596641195,274766987955,
%T A361878 8005895472543,236615835243329,7076435929811769,213755697648537567,
%U A361878 6512143129366530853,199862758637494411349,6173557491107989995435,191779157650960532459435,5987596175475052883532955
%N A361878 a(n) = hypergeom([-n, -n, n, n + 1], [1, 1, 1], 1).
%D A361878 0
%F A361878 From _Peter Bala_, Mar 29 2023: (Start)
%F A361878 a(n) = Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)*binomial(n+k-1,k).
%F A361878 a(n) = (1/12)*(7*A005259(n) + A005259(n-1)) for n >= 1.
%F A361878 P-recursive: n^3*Q(n-1)*a(n) = 4*(204*n^6 - 1275*n^5 + 3178*n^4 - 3999*n^3 + 2667*n^2 - 910*n + 126)*a(n-1) - (n - 2)^3*Q(n)*a(n-2) with a(0) = 1, a(1) = 3 and where Q(n) = 24*n^3 - 42*n^2 + 28*n - 7.
%F A361878 a(n) ~ (1 + sqrt(2))^(4*n+1) / (2^(7/4)*(Pi*n)^(3/2)).
%F A361878 The supercongruence a(n*p^r) == a(n*p^(r-1)) holds for positive integers n and r and all primes p >= 5. (End)
%p A361878 A361878 := n -> hypergeom([-n, -n, n, n + 1], [1, 1, 1], 1):
%p A361878 seq(simplify(A361878(n)), n = 0..18);
%t A361878 Table[HypergeometricPFQ[{-n, -n, n, n + 1}, {1, 1, 1}, 1], {n,0,20}] (* _Vaclav Kotesovec_, Mar 29 2023 *)
%Y A361878 Cf. A005259, A361712.
%K A361878 nonn
%O A361878 0,2
%A A361878 _Peter Luschny_, Mar 27 2023