A116404 Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).

1, 1, 2, 6, 15, 35, 84, 204, 493, 1189, 2870, 6930, 16731, 40391, 97512, 235416, 568345, 1372105, 3312554, 7997214, 19306983, 46611179, 112529340, 271669860, 655869061, 1583407981, 3822685022, 9228778026, 22280241075, 53789260175

Offset: 0

Author

Paul Barry, Feb 07 2006

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A114123.

Binomial transform of A114122.

Congruent to 1,1,0,0,1,1,... modulo 2.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vincent Vatter, Growth rates of permutation classes: from countable to uncountable, arXiv:1605.04297 [math.CO], 2016.
• Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).

Crossrefs

Cf. A000129, A056594, A114122, A114123.

Programs

Magma

I:=[1,1,2,6]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-3)+Self(n-4): n in [1..36]]; // Vincenzo Librandi, Aug 19 2017

Mathematica

LinearRecurrence[{2,0,2,1}, {1,1,2,6}, 35] (* Emanuele Munarini, Apr 27 2017 *)
CoefficientList[Series[(1-x)/((1-x)^2 -x^2(1+x)^2), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2017 *)

PARI

Vec((1-x)/((1-x)^2-x^2*(1+x)^2) + O(x^40)) \\ Michel Marcus, Aug 19 2017

Sage

def A116404(n): return sum( round( hypergeometric([-n+2*k, -2*k], [1], 2) ) for k in (0..n//2) )
[A116404(n) for n in (0..35)] # G. C. Greubel, Nov 20 2021

Formula

G.f.: (1-x)/(1 - 2*x - 2*x^3 - x^4).

a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(2*k,n-2*k-j)*C(n-2*k,j)*2^(n-2*k-j).

2*a(n) = A056594(n) + A000129(n+1). - R. J. Mathar, Oct 25 2011

a(n) = Sum_{k=0..floor(n/2)} hypergeometric2F1([-2*k, -n+2*k], [1], 2). - G. C. Greubel, Nov 20 2021