A098405 Expansion of (1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x))).

1, 7, 47, 327, 2343, 17127, 126951, 950631, 7173991, 54471527, 415652711, 3184708455, 24485137255, 188802730855, 1459525454695, 11307626564455, 87775235181415, 682523302202215, 5315297718995815, 41450938169985895, 323653082644384615, 2529960757626047335

Offset: 0

Author

Paul Barry, Sep 06 2004

Keywords

• easy
• nonn

Comments

Partial sums of A069720.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A069720.

Programs

Magma

[n le 2 select 7^(n-1) else ((9*n-4)*Self(n-1) - 4*(2*n-1)*Self(n-2))/n: n in [1..30]]; // G. C. Greubel, Dec 27 2023

Mathematica

Table[SeriesCoefficient[(1-Sqrt[1-8*x])/((1-x)*(4*x*Sqrt[1-8*x])),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)
Accumulate[Table[2^(n-1) Binomial[2n-1,n],{n,20}]] (* Harvey P. Dale, Jan 20 2013 *)

PARI

my(x='x+O('x^66)); Vec((1-sqrt(1-8*x))/((1-x)*(4*x*sqrt(1-8*x)))) \\ Joerg Arndt, May 11 2013

SageMath

def A098405(n): return sum( binomial(2*k+1,k)*2^k for k in range(n+1))
[A098405(n) for n in range(41)] # G. C. Greubel, Dec 27 2023

Formula

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

D-finite with recurrence: (n+1)*a(n) = (9*n+5)*a(n-1) - 4*(2*n+1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012

a(n) ~ 2^(3*n+4)/(7*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012