A377218 - OEIS

%I #13 Jan 13 2025 20:31:41

%S 1,1,29,2246,239500,30318701,4271201506,647359627557,103476937050223,

%T 17223017775652625,2959285397777331751,521687007046376376544,

%U 93932798602803741121051,17215649571517858590782737,3203146941738318544432065500,603763082812549420389330837978,115095760617137117019641563685386

%N Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(24*n) = (4*n)!/n!^4 for n >= 0.

%C Compare with A000984(n) = [x^n] (1 + x)^(2*n) = (2*n)!/n!^2.

%C The central binomial coefficients A000984(n) satisfy the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) for all primes p >= 5 and positive integers n and k.

%C More generally, for positive integers r and s, the sequence {u(r,s; n) : n >= 0} defined by u(r,s; n) = [x^(s*n)] (1 + x)^(r*n) = binomial(r*n, s*n) satisfies the same supercongruences (Meštrović, Section 6, equation 39).

%C Conjecture: for positive integers r and s, the sequence {v(r,s; n) : n >= 0} defined by v(r,s; n) = [x^(s*n)] A(x)^(r*n) also satisfies the same supercongruences.

%H Romeo Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012)</a>, arXiv:1111.3057 [math.NT], (2011).

%F O.g.f.: A(x) = ( x/(x * series_reversion(E(x)))^(1/24), where E(x) = exp(Sum_{n >= 1} (4*n)!/n!^4 *x^n/n) is the o.g.f. of A333042.

%p Order := 25:

%p E(x) := exp(add((4*n)!/n!^4 * x^n/n, n = 1..25)):

%p solve(series(x*E(x),x) = y, x):

%p convert(%, polynom):

%p g := taylor(y/%, y = 0, 25):

%p seq(coeftayl(g^(1/24), y = 0,  n), n = 0..20);

%Y Cf. A008977, A082368, A333042, A370294, A377217, A377219.

%K nonn,easy

%O 0,3

%A _Peter Bala_, Oct 20 2024