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%I #28 Jul 23 2017 10:24:20
%S 0,0,0,0,1,5,15,35,70,125,200,275,275,0,-1000,-3625,-9500,-21250,
%T -42500,-76875,-124375,-171875,-171875,0,621875,2250000,5890625,
%U 13171875,26343750,47656250,77109375,106562500,106562500,0,-385546875,-1394921875,-3651953125
%N a(n) = Sum_{k>=0} (-1)^k*binomial(n,5*k+4).
%C {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 A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5).
%F G.f.: (-x^4)/((-1+x)^5 - x^5). - _Peter J. C. Moses_, Jul 05 2017
%F For n>=1, a(n) = (2/5)*(phi+2)^(n/2)*(cos(Pi*(n-8)/10) + (phi-1)^n*cos (3* Pi*(n-8)/10)), where phi is the golden ratio;
%F a(n+m) = a(n)*K_1(m) + K_4(n)*K_2(m) + K_3(n)*K_3(m) + K_2(n)*K_4(m) + K_1(n)*a(m), where K_1 is A289306, K_2 is A289321, K_3 is A289387, K_4 is A289388.
%F a(n) = 0 if and only if n=0,1,2 or n==3 (mod 10). - _Vladimir Shevelev_, Jul 15 2017
%t Table[Sum[(-1)^k*Binomial[n, 5 k + 4], {k, 0, n}], {n, 0, 36}] (* or *)
%t CoefficientList[Series[(-x^4)/((-1 + x)^5 - x^5), {x, 0, 36}], x] (* _Michael De Vlieger_, Jul 10 2017 *)
%o (PARI) a(n) = sum(k=0, (n-4)\5, (-1)^k*binomial(n, 5*k+4)); \\ _Michel Marcus_, Jul 05 2017
%Y Cf. A139398, A133476, A139714, A139748, A139761.
%Y Cf. A289306, A289321, A289387, A289388.
%K sign
%O 0,6
%A _Vladimir Shevelev_, Jul 05 2017
%E More terms from _Peter J. C. Moses_, Jul 05 2017
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