OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: hypergeom([1/3, 2/3],[1],27*x^3/(1-x)^2)/(1-x). - Mark van Hoeij, Apr 11 2014
Recurrence: (n-4)*n^2*a(n) = (4*n^3 - 21*n^2 + 23*n - 9)*a(n-1) - (6*n^3 - 39*n^2 + 73*n - 45)*a(n-2) + (31*n^3 - 220*n^2 + 455*n - 279)*a(n-3) - (55*n^3 - 468*n^2 + 1278*n - 1125)*a(n-4) + 3*(n-3)*(3*n - 11)*(3*n - 10)*a(n-5). - Vaclav Kotesovec, Dec 08 2021
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 19*x^4 + 37*x^5 + 151*x^6 + 541*x^7 +...
where
A(x) = 1 +
x*((1-x) + x) +
x^2*((1-x)^2 + 2^3*x*(1-x) + x^2) +
x^3*((1-x)^3 + 3^3*x*(1-x)^2 + 3^3*x^2*(1-x) + x^3) +
x^4*((1-x)^4 + 4^3*x*(1-x)^3 + 6^3*x^2*(1-x)^2 + 4^3*x^3*(1-x) + x^4) +
x^5*((1-x)^5 + 5^3*x*(1-x)^4 + 10^3*x^2*(1-x)^3 + 10^3*x^3*(1-x)^2 + 5^3*x^4*(1-x) + x^5) +...
MATHEMATICA
CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3}, {1}, 27*x^3/(1-x)^2]/(1-x), {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 08 2021 *)
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^3*x^k*(1-x)^(m-k) + x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 19 2012
STATUS
approved