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A377217 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(6*n) = (3*n)!/n!^3 for n >= 0.

%I A377217 #15 Oct 31 2024 01:35:40
%S A377217 1,1,2,14,127,1364,16219,206715,2770342,38567069,553153830,8126285739,
%T A377217 121758839828,1854687918895,28649693078544,447912211497740,
%U A377217 7076246388778874,112821090561117084,1813395701702453669
%N A377217 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(6*n) = (3*n)!/n!^3 for n >= 0.
%C A377217 Compare with A000984(n) = [x^n] (1 + x)^(2*n) = (2*n)!/n!^2.
%C A377217 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 A377217 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 A377217 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 A377217 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 A377217 O.g.f.: A(x) = ( x/(x * series_reversion(E(x)))^(1/6), where E(x) = exp(Sum_{n >= 1} (3*n)!/n!^3 *x^n/n) is the o.g.f. of A229451.
%p A377217 Order := 25:
%p A377217 E(x) := exp(add((3*n)!/n!^3 * x^n/n, n = 1..25)):
%p A377217 solve(series(x*E(x),x) = y, x):
%p A377217 convert(%, polynom):
%p A377217 g := taylor(y/%, y = 0, 25):
%p A377217 seq(coeftayl(g^(1/6), y = 0,  n), n = 0..20);
%Y A377217 Cf. A006480, A060542, A229451, A229452, A377218, A377219.
%K A377217 nonn,easy
%O A377217 0,3
%A A377217 _Peter Bala_, Oct 20 2024