A364305 a(n) = (8*n)!*(5*n)!*(3*n)! / ( (6*n)!*(4*n)!^2*n!^2 ).
1, 70, 17550, 5567380, 1960044750, 732012601320, 283986961467300, 113142133870180800, 45969979122504907470, 18961650930856541865100, 7915377251895103264073800, 3336455614603881320759754000, 1417729131896719482585245182500, 606517077508008639090614765297280
Offset: 0
Links
- Paolo Xausa, Table of n, a(n) for n = 0..300
- R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Programs
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Maple
seq( (8*n)!*(5*n)!*(3*n)! / ( (6*n)!*(4*n)!^2*n!^2 ), n = 0..13);
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Mathematica
A364305[n_]:=(8n)!(5n)!(3n)!/((6n)!(4n)!^2n!^2);Array[A364305,15,0] (* Paolo Xausa, Oct 06 2023 *)
Formula
a(n) = Sum_{k = 0..n} binomial(8*n, n-k)^2 * binomial(6*n+k-1, k).
a(n) = [x^n] (1 - x)^(2*n) * P(8*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial.
a(n) = (5/12)*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*(8*n-1)*(8*n-3)*(8*n-5)*(8*n-7)/((4*n-1)*(4*n-3)*(6*n-1)*(6*n-5)*n^2*(2*n-1)^2) * a(n-1) with a(0) = 1.
a(n) ~ c^n * sqrt(5)/(4*Pi*n), where c = (2^2)*(5^5)/(3^3).
a(p) == a(1) (mod p^3) for all primes p >= 5.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k [added 16 Oct 2024: the conjecture follows from Meštrović, equation 39, since a(n) = binomial(8*n, 2*n)*binomial(5*n, n)* binomial(2*n, n)/binomial(4*n, n)].
a(n) = [x^n] G(x)^(10*n), where the power series G(x) = 1 + 7*x + 412*x^2 + 55524*x^3 + 10088066*x^4 + 2146473322*x^5 + 503731865112*x^6 + ... appears to have integer coefficients.
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x)^10, where the power series F(x) = 1 + 7*x + 902*x^2 + 191779*x^3 + 50706776*x^4 + 15153397742*x^5 + 4898289306180*x^6 + ... appears to have integer coefficients.
From Peter Bala, Oct 16 2024: (Start)
For integer r and positive integer s, define sequences u(n) = { [x^(s*n)] G(x)^(r*n) : n >= 0 } and v(n) = { [x^(s*n)] F(x)^(r*n) : n >= 0 }, with the power series F(x) and G(x) as defined above. We conjecture that both sequences {u(n)} and {v(n)} satisfy the above supercongruences mod p^(3*k). (End)