A375365 Expansion of 1/( (1 + x)^2 * (1 - x^2*(1 + x)^3) ).

1, -2, 4, -3, 6, -2, 14, 3, 32, 35, 92, 142, 309, 541, 1061, 1970, 3770, 7067, 13423, 25328, 47925, 90546, 171268, 323704, 612034, 1157045, 2187523, 4135499, 7818493, 14781207, 27944635, 52830674, 99879267, 188826659, 356986436, 674901081, 1275934925, 2412219595

Offset: 0

Author

Seiichi Manyama, Aug 13 2024

Keywords

• sign

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,0,5,10,10,5,1).

Crossrefs

Cf. A375315, A375317, A375364.

Programs

Mathematica

CoefficientList[Series[1/((1+x)^2(1-x^2(1+x)^3)),{x,0,40}],x] (* or *) LinearRecurrence[{-2,0,5,10,10,5,1},{1,-2,4,-3,6,-2,14},40] (* Harvey P. Dale, Aug 07 2025 *)

PARI

my(N=40, x='x+O('x^N)); Vec(1/((1+x)^2*(1-x^2*(1+x)^3)))

PARI

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

Formula

a(n) = -2*a(n-1) + 5*a(n-3) + 10*a(n-4) + 10*a(n-5) + 5*a(n-6) + a(n-7).

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