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 A289389 a(n) = Sum_{k>=0} (-1)^k*binomial(n,5*k+4). 6
 0, 0, 0, 0, 1, 5, 15, 35, 70, 125, 200, 275, 275, 0, -1000, -3625, -9500, -21250, -42500, -76875, -124375, -171875, -171875, 0, 621875, 2250000, 5890625, 13171875, 26343750, 47656250, 77109375, 106562500, 106562500, 0, -385546875, -1394921875, -3651953125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS {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. REFERENCES A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII. LINKS Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017. Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5). FORMULA G.f.: (-x^4)/((-1+x)^5 - x^5). - Peter J. C. Moses, Jul 05 2017 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; 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. a(n) = 0 if and only if n=0,1,2 or n==3 (mod 10). - Vladimir Shevelev, Jul 15 2017 MATHEMATICA Table[Sum[(-1)^k*Binomial[n, 5 k + 4], {k, 0, n}], {n, 0, 36}] (* or *) CoefficientList[Series[(-x^4)/((-1 + x)^5 - x^5), {x, 0, 36}], x] (* Michael De Vlieger, Jul 10 2017 *) PROG (PARI) a(n) = sum(k=0, (n-4)\5, (-1)^k*binomial(n, 5*k+4)); \\ Michel Marcus, Jul 05 2017 CROSSREFS Cf. A139398, A133476, A139714, A139748, A139761. Cf. A289306, A289321, A289387, A289388. Sequence in context: A292103 A290447 A000750 * A008487 A000743 A138779 Adjacent sequences:  A289386 A289387 A289388 * A289390 A289391 A289392 KEYWORD sign AUTHOR Vladimir Shevelev, Jul 05 2017 EXTENSIONS More terms from Peter J. C. Moses, Jul 05 2017 STATUS approved

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Last modified August 6 15:44 EDT 2020. Contains 336254 sequences. (Running on oeis4.)