OFFSET
0,7
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. - Vladimir Shevelev, Jul 24 2017
REFERENCES
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..3000
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
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.: -((-1+x)^4/((-1+x)^5-x^5)). - Peter J. C. Moses, Jul 02 2017
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).
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
MATHEMATICA
Table[Sum[(-1)^k*Binomial[n, 5 k], {k, 0, n}], {n, 0, 36}] (* or *)
CoefficientList[Series[-((-1 + x)^4/((-1 + x)^5 - x^5)), {x, 0, 36}], x] (* Michael De Vlieger, Jul 04 2017 *)
LinearRecurrence[{5, -10, 10, -5}, {1, 1, 1, 1, 1}, 40] (* Harvey P. Dale, Dec 23 2018 *)
PROG
(PARI) a(n) = sum(k=0, n\5, (-1)^k*binomial(n, 5*k)); \\ Michel Marcus, Jul 02 2017
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Vladimir Shevelev, Jul 02 2017
STATUS
approved