A382539 Expansion of 1/(1 - x/(1 - 4*x)^(1/2))^2.

1, 2, 7, 28, 117, 498, 2139, 9232, 39953, 173162, 751103, 3259132, 14142973, 61367542, 266223083, 1154592752, 5005724185, 21694354406, 93985418399, 407009142836, 1761880487509, 7623911365210, 32976925264827, 142585750821408, 616281411472257, 2662702949358158

Offset: 0

Author

Seiichi Manyama, Mar 31 2025

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A026671, A382540.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(1/2))^2)); // Vincenzo Librandi, May 13 2025

Mathematica

Table[Sum[4^(n-k)* (k+1)* Binomial[n-k/2-1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 13 2025 *)

PARI

a(n) = sum(k=0, n, 4^(n-k)*(k+1)*binomial(n-k/2-1, n-k));

Formula

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

D-finite with recurrence (-n+1)*a(n) +2*(4*n-7)*a(n-1) +(-15*n+41)*a(n-2) +2*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Apr 02 2025