|
|
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
|
|
|
FORMULA
|
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.
|
|
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
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|