cp's OEIS Frontend

A208426 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-3*x)^(3*n+1).

%I A208426 #22 Jul 05 2025 09:56:31
%S A208426 1,3,15,99,711,5373,42099,338355,2771127,23028813,193610385,
%T A208426 1643215005,14056350075,121040308665,1048212778635,9122168556819,
%U A208426 79727173530327,699443806767525,6156776010386481,54356715121718349,481194980656865721,4270165015550478003
%N A208426 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-3*x)^(3*n+1).
%C A208426 Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
%C A208426 Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 3*x*y*z), 1/(1 - x*y + y*z + x*z - 3*x*y*z). - _Gheorghe Coserea_, Jul 04 2018
%C A208426 Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + 3*x*y*z)). - _Seiichi Manyama_, Jul 05 2025
%H A208426 Gheorghe Coserea, <a href="/A208426/b208426.txt">Table of n, a(n) for n = 0..200</a>
%F A208426 From _Gheorghe Coserea_, Jul 04 2018: (Start)
%F A208426 a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k).
%F A208426 G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 9*x - 1)*y'' + (243*x^4 + 216*x^3 + 27*x^2 + 36*x - 2)*y' + 3*(27*x^3 + 33*x^2 - 2*x + 2)*y.
%F A208426 (End)
%F A208426 From _Vaclav Kotesovec_, Oct 07 2020: (Start)
%F A208426 Recurrence: n^2*(3*n - 5)*a(n) = 3*(9*n^3 - 24*n^2 + 17*n - 4)*a(n-1) + 3*(3*n - 4)*a(n-2) + 27*(n-2)^2*(3*n - 2)*a(n-3).
%F A208426 a(n) ~ sqrt(2 + sqrt(5)*phi^(-1/3) + sqrt(5)*phi^(1/3)) * 3^n * (1 + phi^(-2/3) + phi^(2/3))^n / (2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
%F A208426 (End)
%e A208426 G.f.: A(x) = 1 + 3*x + 15*x^2 + 99*x^3 + 711*x^4 + 5373*x^5 + 42099*x^6 + ...
%e A208426 where
%e A208426 A(x) = 1/(1-3*x) + 6*x^2/(1-3*x)^4 + 90*x^4/(1-3*x)^7 + 1680*x^6/(1-3*x)^10 + 34650*x^8/(1-3*x)^13 + 756756*x^10/(1-3*x)^16 + ...
%t A208426 Table[3^n * HypergeometricPFQ[{1/2 - n/2, -n/2, 1 + n}, {1, 1}, 4/9], {n, 0, 25}] (* _Vaclav Kotesovec_, Oct 07 2020 *)
%o A208426 (PARI) {a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-3*x+x*O(x^n))^(3*m+1)),n)}
%o A208426 for(n=0,31,print1(a(n),", "))
%o A208426 (PARI) a(n) = sum(k=0, n\2, (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k)); \\ _Gheorghe Coserea_, Jul 04 2018
%Y A208426 Cf. A000172, A002893, A069835, A208425, A294035.
%K A208426 nonn
%O A208426 0,2
%A A208426 _Paul D. Hanna_, Feb 26 2012