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A362733 a(n) = [x^n] F(x)^n, where F(x) = exp( Sum_{k >= 1} A362732(k)*x^k/k ).

%I A362733 #14 Oct 31 2024 01:35:09
%S A362733 1,6,234,10428,492522,24033006,1197423396,60530725380,3092592004074,
%T A362733 159295600885794,8258018380659234,430335300869496072,
%U A362733 22521831447746893092,1182951246247357578348,62325193477833011143260,3292376206935392483917428,174323297281680647978503146,9248680725006429075147528150
%N A362733 a(n) = [x^n] F(x)^n, where F(x) = exp( Sum_{k >= 1} A362732(k)*x^k/k ).
%F A362733 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 A362733 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 A362733 For n >= 1, a(n) = (1/2) * [x^n] G(x)^(2*n), where G(x) = exp(Sum_{k >= 1} A006480(k)*x^k/k) is the g.f. of A229451. - _Peter Bala_, Oct 27 2024
%p A362733 E(n,x) := series(exp(n*add((3*k)!/k!^3*x^k/k, k = 1..20)), x, 21):
%p A362733 A362732(n) := coeftayl(E(n,x), x = 0, n):
%p A362733 F(n,x) := series(exp(n*add(A362732(k)*x^k/k, k = 1..20)), x, 21):
%p A362733 seq(coeftayl(F(n,x), x = 0, n), n = 0..20);
%p A362733 # alternative program
%p A362733 G(n,x) := series(exp(n*add((3*k)!/k!^3*x^(2*k)/k, k = 1..40)), x, 41):
%p A362733 seq((1/2)*coeftayl(G(2*n,x), x = 0, 2*n), n = 1..20); # _Peter Bala_, Oct 27 2024
%Y A362733 Cf. A006480, A229451, A362722 - A362732.
%K A362733 nonn,easy
%O A362733 0,2
%A A362733 _Peter Bala_, May 06 2023