This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A289306 #35 Mar 21 2019 07:11:03
%S A289306 1,1,1,1,1,0,-5,-20,-55,-125,-250,-450,-725,-1000,-1000,0,3625,13125,
%T A289306 34375,76875,153750,278125,450000,621875,621875,0,-2250000,-8140625,
%U A289306 -21312500,-47656250,-95312500,-172421875,-278984375,-385546875,-385546875,0,1394921875
%N A289306 a(n) = Sum_{k >= 0}(-1)^k*binomial(n,5*k).
%C A289306 {A289306, A289321, A289387, A289388, A289389} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x), k_5(x)} of order 5. For the definitions of {k_i(x)} and the difference analog {K_i (n)} see [Erdelyi] and the Shevelev link respectively. - _Vladimir Shevelev_, Jul 24 2017
%D A289306 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H A289306 Seiichi Manyama, <a href="/A289306/b289306.txt">Table of n, a(n) for n = 0..3000</a>
%H A289306 John B. Dobson, <a href="http://arxiv.org/abs/1610.09361">A matrix variation on Ramus's identity for lacunary sums of binomial coefficients</a>, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
%H A289306 Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.
%H A289306 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5).
%F A289306 G.f.: -((-1+x)^4/((-1+x)^5-x^5)). - _Peter J. C. Moses_, Jul 02 2017
%F A289306 For n>=1, a(n) = (2/5)*(phi+2)^(n/2)*(cos(Pi*n/10) + (phi-1)^n*cos(3 * Pi* n/10)), where phi is the golden ratio. In particular, a(n) = 0 if and only if n==5 (mod 10).
%F A289306 a(n+m) = a(n)*a(m) - K_5(n)*K_2(m) - K_4(n)*K_3(m) - K_3(n)*K_4(m) - K_2(n)*K_5(m), where K_2 is A289321, K_3 is A289387, K_4 is A289388, K_5 is A289389. - _Vladimir Shevelev_, Jul 24 2017
%t A289306 Table[Sum[(-1)^k*Binomial[n, 5 k], {k, 0, n}], {n, 0, 36}] (* or *)
%t A289306 CoefficientList[Series[-((-1 + x)^4/((-1 + x)^5 - x^5)), {x, 0, 36}], x] (* _Michael De Vlieger_, Jul 04 2017 *)
%t A289306 LinearRecurrence[{5,-10,10,-5},{1,1,1,1,1},40] (* _Harvey P. Dale_, Dec 23 2018 *)
%o A289306 (PARI) a(n) = sum(k=0, n\5, (-1)^k*binomial(n,5*k)); \\ _Michel Marcus_, Jul 02 2017
%Y A289306 Column 5 of A307039.
%Y A289306 Cf. A139398, A133476, A139714, A139748, A139761.
%K A289306 sign,easy
%O A289306 0,7
%A A289306 _Vladimir Shevelev_, Jul 02 2017