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A116404
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Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).
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4
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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
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OFFSET
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0,3
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COMMENTS
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Diagonal sums of number triangle A114123.
Congruent to 1,1,0,0,1,1,... modulo 2.
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LINKS
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FORMULA
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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).
a(n) = Sum_{k=0..floor(n/2)} hypergeometric2F1([-2*k, -n+2*k], [1], 2). - G. C. Greubel, Nov 20 2021
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MATHEMATICA
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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 *)
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PROG
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(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
(Sage)
def A116404(n): return sum( round( hypergeometric([-n+2*k, -2*k], [1], 2) ) for k in (0..n//2) )
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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