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A318108 a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-3)^k, n >= 0.

%I A318108 #33 Nov 28 2024 15:22:29
%S A318108 1,3,27,303,3771,49653,677979,9496791,135572859,1963940073,
%T A318108 28783474677,425872190241,6350923156059,95341185353781,
%U A318108 1439433069482547,21839152342265703,332769145298428539,5089688869615075521,78108038975852093889,1202268428203687094493,18555675891246972931221
%N A318108 a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-3)^k, n >= 0.
%C A318108 Diagonal of rational function 1/(1 - (x + y + z - 3*x*y*z)).
%H A318108 Seiichi Manyama, <a href="/A318108/b318108.txt">Table of n, a(n) for n = 0..829</a> (terms 0..100 from Gheorghe Coserea)
%F A318108 G.f. y=A(x) satisfies: 0 = x*(6*x - 1)*(27*x^3 + 27*x^2 - 18*x + 1)*y'' + (486*x^4 + 216*x^3 - 189*x^2 + 36*x - 1)*y' + 3*(3*x + 1)*(18*x^2 - 6*x + 1)*y.
%F A318108 Recurrence: n^2*(3*n - 4)*a(n) = 3*(3*n - 2)*(6*n^2 - 10*n + 3)*a(n-1) - 9*(9*n^3 - 30*n^2 + 29*n - 6)*a(n-2) - 27*(n-2)^2*(3*n - 1)*a(n-3). - _Vaclav Kotesovec_, Mar 01 2019
%F A318108 From _Peter Bala_, Mar 16 2023: (Start)
%F A318108 a(n) = Sum_{k = 0..n} (-3)^(n-k)*binomial(n,k)*binomial(n + 2*k,n)* binomial(2*k,k).
%F A318108 Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. (End)
%F A318108 G.f.: hypergeom([1/3, 2/3],[1],27*x/(1+3*x)^3)/(1+3*x). - _Mark van Hoeij_, Nov 28 2024
%e A318108 A(x) = 1 + 3*x + 27*x^2 + 303*x^3 + 3771*x^4 + 49653*x^5 + 677979*x^6 + ...
%t A318108 Table[Sum[(3n-2k)!/(((n-k)!)^3 k!) (-3)^k,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Mar 01 2019 *)
%o A318108 (PARI)
%o A318108 a(n) = sum(k=0, n, (3*n-2*k)!/((n-k)!^3*k!)*(-3)^k);
%o A318108 vector(21, n, a(n-1))
%Y A318108 Cf. A000172, A124435, A318107, A318109.
%K A318108 nonn,easy
%O A318108 0,2
%A A318108 _Gheorghe Coserea_, Sep 20 2018