A383503 Expansion of 1/sqrt( (1-x) * (1-x-4*x^3)^3 ).

1, 2, 3, 10, 23, 42, 97, 218, 435, 918, 1977, 4062, 8393, 17590, 36303, 74614, 154211, 317334, 650505, 1335054, 2736453, 5595950, 11439475, 23370270, 47681965, 97217882, 198110199, 403383026, 820820215, 1669405626, 3393344257, 6893850650, 13999109715, 28414742790

Offset: 0

Author

Seiichi Manyama, May 05 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A383254, A383499.

Cf. A217615, A377213.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/Sqrt( (1-x) * (1-x-4*x^3)^3 ))); // Vincenzo Librandi, May 06 2025

Mathematica

Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-2*k+1,k+1],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

PARI

a(n) = sum(k=0, n\3, (2*k+1)*binomial(2*k, k)*binomial(n-2*k+1, k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} (2*k+1) * binomial(2*k,k) * binomial(n-2*k+1,k+1).