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URL: https://oeis.org/A116404

⇱ A116404 - OEIS

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A116404
Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).
4
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, 129858761424, 313506783024, 756872327473
OFFSET
0,3
COMMENTS
Diagonal sums of number triangle A114123.
Binomial transform of A114122.
Congruent to 1,1,0,0,1,1,... modulo 2.
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
E.g.f.: (cos(x) + exp(x)*cosh(sqrt(2)*x) + exp(x)*sinh(sqrt(2)*x)/sqrt(2)) / 2. - Amiram Eldar, Feb 19 2026
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 *)
PROG
(PARI) Vec((1-x)/((1-x)^2-x^2*(1+x)^2) + O(x^40)) \\ Michel Marcus, Aug 19 2017
(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
(SageMath)
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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 07 2006
STATUS
approved