A364244 a(n) = A143007(2*n-1, n-1) for n >= 1.

1, 25, 1441, 107353, 9073501, 826861993, 79219824685, 7865844936025, 802198564524325, 83532710607121525, 8844234718023010681, 949244022625120188265, 103044177225432902852641, 11293765432962617876667253, 1248038875078327818254657941

Offset

1,2

Comments

The sequence of Apéry numbers A005259 forms the main diagonal of A143007, i.e., A005259(n) = A143007(n, n). The Apéry numbers satisfy the supercongruences A005259(n*p^r) == A005259(n^p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r. We conjecture that the present sequence satisfies the same supercongruences.

More generally, for positive integers r and s, the sequence defined by a(r,s;n) = A143007(r*n - 1, s*n - 1) may also satisfy the same supercongruences. This is the case r = 2, s = 1. Compare with the comments in A363864.

Links

Table of n, a(n) for n=1..15.

Formula

a(n) = Sum_{k = 0..n-1} binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2.

a(n) = hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1).

P-recursive: 2*(n-1)^3*(2*n-1)^3*(440*n^3-2178*n^2+3600*n-1987)*a(n) = (865920*n^9 - 9481824*n^8 + 45492136*n^7 - 125359294*n^6 + 218361816*n^5 - 249018285*n^4 + 185709390*n^3 - 87271191*n^2 + 23447876*n - 2745998)*a(n-1) - 2*(2*n-3)^3*(n-2)^3*(440*n^3-858*n^2+564*n-125)*a(n-2) with a(1) = 1 and a(2) = 25.

a(n) ~ phi^(10*n - 4) / (2^(5/2) * 5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 16 2023

Maple

seq( add(binomial(2*n-1, k)^2 * binomial(3*n-2-k, 2*n-1)^2, k = 0..n-1), n = 1..20);
# alternative program
seq(simplify(hypergeom([2*n, 1 - 2*n, n, 1 - n], [1, 1, 1], 1)), n = 1..20);

Mathematica

Table[HypergeometricPFQ[{2*n, 1 - 2*n, n, 1 - n}, {1, 1, 1}, 1], {n, 1, 20}] (* Vaclav Kotesovec, Jul 16 2023 *)

Crossrefs

Cf. A005259, A143007, A363864, A364243.

Sequence in context: A243943 A192107 A206465 * A138246 A219091 A196299

Adjacent sequences: A364241 A364242 A364243 * A364245 A364246 A364247

Keyword

nonn,easy

Author

Peter Bala, Jul 16 2023

Extensions

Offset changed by Georg Fischer, Nov 03 2023

Status

approved