A387208 Expansion of sqrt((1-x) / (1-9*x)^3).

1, 13, 145, 1517, 15329, 151565, 1476465, 14228205, 135990465, 1291409165, 12199991633, 114761111789, 1075651464865, 10051341904141, 93677905064497, 871083359663085, 8083754402585985, 74885500462111245, 692624008942816785, 6397104350057979885, 59008673876627412321

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

n*a(n) = (10*n+3)*a(n-1) - 9*(n-1)*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)/(1-2*(n-k)).

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

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

a(n) ~ 2^(5/2) * sqrt(n) * 3^(2*n-1) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025

Mathematica

CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^3], {x, 0, 33}], x] (* Vincenzo Librandi, Aug 23 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-9*x)^3))
(Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := Sqrt((1-x) / (1-9*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

Crossrefs

Cf. A163869, A163872, A387210.

Cf. A085363, A383946.

Sequence in context: A038492 A270579 A297223 * A199023 A152585 A014881

Adjacent sequences: A387205 A387206 A387207 * A387209 A387210 A387211

Keyword

nonn

Author

Seiichi Manyama, Aug 22 2025

Status

approved