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A289306 a(n) = Sum_{k >= 0}(-1)^k*binomial(n,5*k). 7
1, 1, 1, 1, 1, 0, -5, -20, -55, -125, -250, -450, -725, -1000, -1000, 0, 3625, 13125, 34375, 76875, 153750, 278125, 450000, 621875, 621875, 0, -2250000, -8140625, -21312500, -47656250, -95312500, -172421875, -278984375, -385546875, -385546875, 0, 1394921875 (list; graph; refs; listen; history; text; internal format)
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
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
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
Column 5 of A307039.
Sequence in context: A090133 A351931 A366057 * A325731 A348310 A062988
KEYWORD
sign,easy
AUTHOR
Vladimir Shevelev, Jul 02 2017
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)