A362733 - OEIS

%I #14 Oct 31 2024 01:35:09

%S 1,6,234,10428,492522,24033006,1197423396,60530725380,3092592004074,

%T 159295600885794,8258018380659234,430335300869496072,

%U 22521831447746893092,1182951246247357578348,62325193477833011143260,3292376206935392483917428,174323297281680647978503146,9248680725006429075147528150

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

%F 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 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 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 E(n,x) := series(exp(n*add((3*k)!/k!^3*x^k/k, k = 1..20)), x, 21):

%p A362732(n) := coeftayl(E(n,x), x = 0, n):

%p F(n,x) := series(exp(n*add(A362732(k)*x^k/k, k = 1..20)), x, 21):

%p seq(coeftayl(F(n,x), x = 0, n), n = 0..20);

%p # alternative program

%p G(n,x) := series(exp(n*add((3*k)!/k!^3*x^(2*k)/k, k = 1..40)), x, 41):

%p seq((1/2)*coeftayl(G(2*n,x), x = 0, 2*n), n = 1..20); # _Peter Bala_, Oct 27 2024

%Y Cf. A006480, A229451, A362722 - A362732.

%K nonn,easy

%O 0,2

%A _Peter Bala_, May 06 2023