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

1, 3, 15, 99, 711, 5373, 42099, 338355, 2771127, 23028813, 193610385, 1643215005, 14056350075, 121040308665, 1048212778635, 9122168556819, 79727173530327, 699443806767525, 6156776010386481, 54356715121718349, 481194980656865721, 4270165015550478003

Offset

0,2

Comments

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).

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

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..200

Formula

From Gheorghe Coserea, Jul 04 2018: (Start)

a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k).

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.

(End)

From Vaclav Kotesovec, Oct 07 2020: (Start)

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).

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.

(End)

Example

G.f.: A(x) = 1 + 3*x + 15*x^2 + 99*x^3 + 711*x^4 + 5373*x^5 + 42099*x^6 + ...
where
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 + ...

Mathematica

Table[3^n * HypergeometricPFQ[{1/2 - n/2, -n/2, 1 + n}, {1, 1}, 4/9], {n, 0, 25}] (* Vaclav Kotesovec, Oct 07 2020 *)

Prog

(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)}
for(n=0, 31, print1(a(n), ", "))
(PARI) a(n) = sum(k=0, n\2, (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k)); \\ Gheorghe Coserea, Jul 04 2018

Crossrefs

Cf. A000172, A002893, A208425.

Sequence in context: A140286 A199416 A046635 * A168344 A091713 A156106

Adjacent sequences: A208423 A208424 A208425 * A208427 A208428 A208429

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2012

Status

approved