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A362732 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A006480(k)*x^k/k ).

%I A362732 #27 Jan 27 2025 14:28:06
%S A362732 1,6,162,5082,170274,5920506,210808494,7631158674,279617726754,
%T A362732 10341283241130,385275082939662,14439312879759378,543815325940475694,
%U A362732 20565700004741265900,780470358196543271622,29708379800729905316832,1133811403010621704628514,43371319655978568356324868
%N A362732 a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A006480(k)*x^k/k ).
%C A362732 The de Bruijn numbers A006480(n) = (3*n)!/n!^3 satisfy the supercongruences A006480(n*p^r) == A006480(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and positive integers n and r (write a(n) as binomial(3*n,2*n)*binomial(2*n,n) and apply Mestrovic, equation 39, p. 12).
%C A362732 We inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A006480(n) and, for i >= 1, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. The present sequence is {a(1,n)}. See A362733 for {a(2,n)}.
%C A362732 We conjecture that the sequences {a(i,n) : n >= 0}, i >= 1, satisfy the same supercongruences as above.
%C A362732 For i >= 1, a(i, n) = 1/i *[x^n] E(x)^(i*n), where E(x) = exp(Sum_{k >= 1} (3*k)!/k!^3 * x^k/k) is the g.f. of A229451. - _Peter Bala_, Oct 30 2024
%H A362732 Vaclav Kotesovec, <a href="/A362732/b362732.txt">Table of n, a(n) for n = 0..620</a>
%H A362732 Romeo Meštrović, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%F A362732 a(n*p^r) == a(n*p^(r-1)) (mod p^r) (Gauss congruence) holds for all primes p and positive integers n and r.
%F A362732 Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 3 and positive integers n and r.
%F A362732 a(n) ~ c * d^n / sqrt(n), where d = 39.5963012687026... and c = 0.107979249748...
%p A362732 E(n,x) := series(exp(n*add(((3*k)!/k!^3*x^k)/k, k = 1..20)), x, 21):
%p A362732 seq(coeftayl(E(n,x), x = 0, n), n = 0..20);
%t A362732 nmax = 20; Table[SeriesCoefficient[E^(n*Sum[(3*k)!/k!^3*x^k/k, {k, 1, n}]), {x, 0, n}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Nov 26 2024 *)
%o A362732 (Python)
%o A362732 from sympy import symbols, factorial, exp, series
%o A362732 x = symbols('x')
%o A362732 nmax = 10
%o A362732 result = []
%o A362732 for n in range(0, nmax + 1):
%o A362732     summation = sum(factorial(3 * k) / (factorial(k)**3 * k) * x**k for k in range(1, n + 1))
%o A362732     exp_series = exp(n * summation)
%o A362732     coefficient = exp_series.series(x, 0, n + 1).coeff(x, n)
%o A362732     result.append(coefficient)
%o A362732 print(result) # _Robert C. Lyons_, Jan 27 2025
%Y A362732 Cf. A006480, A229451, A362722 - A362733.
%K A362732 nonn,easy
%O A362732 0,2
%A A362732 _Peter Bala_, May 06 2023