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 A289388 #23 Jul 23 2017 10:26:24
%S A289388 0,0,0,1,4,10,20,35,55,75,75,0,-275,-1000,-2625,-5875,-11750,-21250,
%T A289388 -34375,-47500,-47500,0,171875,621875,1628125,3640625,7281250,
%U A289388 13171875,21312500,29453125,29453125,0,-106562500,-385546875,-1009375000,-2257031250,-4514062500
%N A289388 a(n) = Sum_{k>=0} (-1)^k*binomial(n,5*k+3).
%C A289388 {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.
%D A289388 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H A289388 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 A289388 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5).
%F A289388 G.f.: ((-1+x)*x^3)/((-1+x)^5 - x^5). - _Peter J. C. Moses_, Jul 05 2017
%F A289388 For n>=1, a(n) = (2/5)*(phi+2)^(n/2)*(cos(Pi*(n-6)/10) + (phi-1)^n*cos (3* Pi*(n-6)/10)), where phi is the golden ratio.
%F A289388 a(n+m) = a(n)*K_1(m) + K_3(n)*K_2(m) + K_2(n)*K_3(m) + K_1(n)*a(m) - K_5(n)*K_5(m), where K_1 is A289306, K_2 is A289321, K_3 is A289387, K_5 is A289389.
%F A289388 a(n) = 0 if and only if n=0, n=2 or n==1 (mod 10). - _Vladimir Shevelev_, Jul 15 2017
%t A289388 Table[Sum[(-1)^k*Binomial[n, 5 k + 3], {k, 0, n}], {n, 0, 36}] (* or *)
%t A289388 CoefficientList[Series[((-1 + x) x^3)/((-1 + x)^5 - x^5), {x, 0, 36}], x] (* _Michael De Vlieger_, Jul 10 2017 *)
%o A289388 (PARI) a(n) = sum(k=0, (n-3)\5, (-1)^k*binomial(n, 5*k+3)); \\ _Michel Marcus_, Jul 05 2017
%Y A289388 Cf. A139398, A133476, A139714, A139748, A139761.
%Y A289388 Cf. A289306, A289321, A289387, A289389.
%K A289388 sign
%O A289388 0,5
%A A289388 _Vladimir Shevelev_, Jul 05 2017
%E A289388 More terms from _Peter J. C. Moses_, Jul 05 2017