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%I #23 Dec 25 2022 11:48:34
%S 0,1,2,3,4,5,5,0,-20,-75,-200,-450,-900,-1625,-2625,-3625,-3625,0,
%T 13125,47500,124375,278125,556250,1006250,1628125,2250000,2250000,0,
%U -8140625,-29453125,-77109375,-172421875,-344843750,-623828125,-1009375000,-1394921875
%N a(n) = Sum_{k >= 0}(-1)^k*binomial(n,5*k+1).
%C a(n) = 0 for n == 7 (mod 10). - _Robert Israel_, Jul 12 2017
%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. - _Vladimir Shevelev_, Jul 24 2017
%H Robert Israel, <a href="/A289321/b289321.txt">Table of n, a(n) for n = 0..3579</a>
%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.: ((-1+x)^3 x)/((-1+x)^5-x^5). - _Peter J. C. Moses_, Jul 02 2017
%F For n>=1, a(n) = (2/5)*(phi+2)^(n/2)*(cos(Pi*(n-2)/10) + (phi-1)^n* cos (3*Pi*(n-2)/10)), where phi is the golden ratio.
%F a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_5(n)*K_3(m) - K_4(n)*K_4(m) - K_3(n)*K_5(m), where K_1 is A289306, K_3 is A289387, K_4 is A289388, K_5 is A289389. - _Vladimir Shevelev_, Jul 24 2017
%p f:= gfun:-rectoproc({5*a(n)-10*a(n+1)+10*a(n+2)-5*a(n+3)+a(n+4), a(0)=0,
%p a(1)=1, a(2)=2, a(3) = 3, a(4)=4}, a(n), remember):
%p map(f, [$0..40]); # _Robert Israel_, Jul 11 2017
%t Table[Sum[(-1)^k*Binomial[n, 5 k + 1], {k, 0, n}], {n, 0, 35}] (* or *)
%t CoefficientList[Series[((-1 + x)^3 x)/((-1 + x)^5 - x^5), {x, 0, 35}], x] (* _Michael De Vlieger_, Jul 04 2017 *)
%t LinearRecurrence[{5,-10,10,-5},{0,1,2,3,4},40] (* _Harvey P. Dale_, Dec 25 2022 *)
%o (PARI) a(n) = sum(k=0, (n-1)\5, (-1)^k*binomial(n, 5*k+1)); \\ _Michel Marcus_, Jul 03 2017
%Y Cf. A139398, A133476, A139714, A139748, A139761, A289306.
%K sign
%O 0,3
%A _Vladimir Shevelev_, Jul 02 2017
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