A383947 Expansion of 1/sqrt((1+x)^3 * (1-7*x)).

1, 2, 15, 84, 525, 3318, 21371, 139240, 915417, 6060330, 40345767, 269825724, 1811432805, 12200012958, 82394389395, 557794589904, 3784079617713, 25718668160850, 175085306697791, 1193682452744740, 8148955372804029, 55697327430265158, 381099865385716395

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• changed

Links

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

Crossrefs

Cf. A085458, A322242.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1+x)^3 * (1-7*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1+x)^3*(1-7*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

PARI

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

Formula

n*a(n) = (6*n-4)*a(n-1) + 7*n*a(n-2) for n > 1.

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

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

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