A217421 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^k*(1-x)^(n-k).

1, 1, 1, 7, 19, 37, 151, 541, 1477, 4999, 18277, 58651, 195001, 694747, 2380429, 8091223, 28449667, 99709057, 346212319, 1217144833, 4300487251, 15137730889, 53484426379, 189831062953, 673630230931, 2393381462941, 8527594750507, 30416413412917, 108577772050459

Offset: 0

Author

Paul D. Hanna, Oct 19 2012

Keywords

• nonn

Examples

			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) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A217615.

Programs

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 *)

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), ", "))

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