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

1, 3, 43, 849, 19371, 480503, 12587065, 342634365, 9596641195, 274766987955, 8005895472543, 236615835243329, 7076435929811769, 213755697648537567, 6512143129366530853, 199862758637494411349, 6173557491107989995435, 191779157650960532459435, 5987596175475052883532955

Offset: 0

Author

Peter Luschny, Mar 27 2023

Keywords

• nonn

References

• 0

Crossrefs

Cf. A005259, A361712.

Programs

Maple

A361878 := n -> hypergeom([-n, -n, n, n + 1], [1, 1, 1], 1):
seq(simplify(A361878(n)), n = 0..18);

Mathematica

Table[HypergeometricPFQ[{-n, -n, n, n + 1}, {1, 1, 1}, 1], {n,0,20}] (* Vaclav Kotesovec, Mar 29 2023 *)

Formula

From Peter Bala, Mar 29 2023: (Start)

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

a(n) = (1/12)*(7*A005259(n) + A005259(n-1)) for n >= 1.

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.

a(n) ~ (1 + sqrt(2))^(4*n+1) / (2^(7/4)*(Pi*n)^(3/2)).

The supercongruence a(n*p^r) == a(n*p^(r-1)) holds for positive integers n and r and all primes p >= 5. (End)