A383946 Expansion of 1/sqrt((1-9*x)^3 * (1-x)).

1, 14, 159, 1676, 17005, 168570, 1645035, 15873240, 151863705, 1443272870, 13643264503, 128404376292, 1204055841157, 11255397745298, 104933302809795, 976016662472880, 9059771065058865, 83945271527170110, 776569280469986895, 7173673630527966780, 66182347507155379101, 609866573826736447914

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A084771, A331516.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 9*x)^3 * (1-x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1-9*x)^3*(1-x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-9*x)^3*(1-x)))

Formula

a(n) = A331516(n) - A331516(n-1).

n*a(n) = (10*n+4)*a(n-1) - 9*n*a(n-2) for n > 1.

a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).

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

a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).