A387212 Expansion of sqrt((1-3*x) / (1-7*x)^3).

1, 9, 75, 599, 4659, 35595, 268485, 2005785, 14873715, 109643195, 804354417, 5877232773, 42798735805, 310767250773, 2250899498763, 16267896905895, 117347641620435, 845043416086635, 6076092412278465, 43629213402099045, 312892629725930121, 2241442380182752209

Offset: 0

Author

Seiichi Manyama, Aug 22 2025

Keywords

• nonn
• new

Links

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

Crossrefs

Cf. A163869, A387211.

Cf. A360318, A385813.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 3*x) / (1-7*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

Mathematica

CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)^3))

Formula

n*a(n) = (10*n-1)*a(n-1) - 21*(n-1)*a(n-2) for n > 1.

a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).

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

a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).