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A377219 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(120*n) = (5*n)!/n!^5 for n >= 0.

%I A377219 #14 Jan 13 2025 20:31:37
%S A377219 1,1,353,318986,408941594,633438203535,1105336091531052,
%T A377219 2093867978990821853,4212168629863126220194,8871676970891643267231886,
%U A377219 19375253437183554713216237582,43574669954100844749472466829032,100404408695672206422230611142618195,236114213302057579962294974098604849352,564982003808755415617353442524468859709030
%N A377219 Expansion of the o.g.f. A(x) defined by [x^n] A(x)^(120*n) = (5*n)!/n!^5 for n >= 0.
%C A377219 Compare with A000984(n) = [x^n] (1 + x)^(2*n) = (2*n)!/n!^2.
%C A377219 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 A377219 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 A377219 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 A377219 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 A377219 O.g.f.: A(x) = ( x/(x * series_reversion(E(x)))^(1/120), where E(x) = exp(Sum_{n >= 1} (5*n)!/n!^5 *x^n/n) is the o.g.f. of A333043.
%p A377219 Order := 25:
%p A377219 E(x) := exp(add((5*n)!/n!^5 * x^n/n, n = 1..25)):
%p A377219 solve(series(x*E(x),x) = y, x):
%p A377219 convert(%, polynom):
%p A377219 g := taylor(y/%, y = 0, 25):
%p A377219 seq(coeftayl(g^(1/120), y = 0,  n), n = 0..20);
%Y A377219 Cf. A008978, A322252, A333043, A370295, A377217, A377218.
%K A377219 nonn,easy
%O A377219 0,3
%A A377219 _Peter Bala_, Oct 20 2024