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A289321
a(n) = Sum_{k >= 0}(-1)^k*binomial(n,5*k+1).
6
0, 1, 2, 3, 4, 5, 5, 0, -20, -75, -200, -450, -900, -1625, -2625, -3625, -3625, 0, 13125, 47500, 124375, 278125, 556250, 1006250, 1628125, 2250000, 2250000, 0, -8140625, -29453125, -77109375, -172421875, -344843750, -623828125, -1009375000, -1394921875
OFFSET
0,3
COMMENTS
a(n) = 0 for n == 7 (mod 10). - Robert Israel, Jul 12 2017
{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
FORMULA
G.f.: ((-1+x)^3 x)/((-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-2)/10) + (phi-1)^n* cos (3*Pi*(n-2)/10)), where phi is the golden ratio.
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
MAPLE
f:= gfun:-rectoproc({5*a(n)-10*a(n+1)+10*a(n+2)-5*a(n+3)+a(n+4), a(0)=0,
a(1)=1, a(2)=2, a(3) = 3, a(4)=4}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Jul 11 2017
MATHEMATICA
Table[Sum[(-1)^k*Binomial[n, 5 k + 1], {k, 0, n}], {n, 0, 35}] (* or *)
CoefficientList[Series[((-1 + x)^3 x)/((-1 + x)^5 - x^5), {x, 0, 35}], x] (* Michael De Vlieger, Jul 04 2017 *)
LinearRecurrence[{5, -10, 10, -5}, {0, 1, 2, 3, 4}, 40] (* Harvey P. Dale, Dec 25 2022 *)
PROG
(PARI) a(n) = sum(k=0, (n-1)\5, (-1)^k*binomial(n, 5*k+1)); \\ Michel Marcus, Jul 03 2017
CROSSREFS
KEYWORD
sign
AUTHOR
Vladimir Shevelev, Jul 02 2017
STATUS
approved