A387233 Expansion of sqrt((1-2*x) / (1-6*x)^5).

1, 14, 142, 1252, 10190, 78724, 586236, 4247688, 30132438, 210175540, 1445920388, 9833940472, 66237449356, 442463439656, 2934485313400, 19340115356688, 126759642351462, 826734451831956, 5368338057048756, 34721155684000920, 223765535492622564, 1437403425873718776

Offset: 0

Author

Seiichi Manyama, Aug 23 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A387228, A387234.

Cf. A360317, A387211.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 2*x) / (1-6*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

Mathematica

CoefficientList[Series[Sqrt[(1-2*x)/(1-6*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)^5))

Formula

n*a(n) = (8*n+6)*a(n-1) - 12*n*a(n-2) for n > 1.

a(n) = (1/2)^n * Sum_{k=0..n} 3^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).

a(n) = Sum_{k=0..n} 2^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).

a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).