A289388 a(n) = Sum_{k>=0} (-1)^k*binomial(n,5*k+3).

0, 0, 0, 1, 4, 10, 20, 35, 55, 75, 75, 0, -275, -1000, -2625, -5875, -11750, -21250, -34375, -47500, -47500, 0, 171875, 621875, 1628125, 3640625, 7281250, 13171875, 21312500, 29453125, 29453125, 0, -106562500, -385546875, -1009375000, -2257031250, -4514062500

Offset: 0

Author

Vladimir Shevelev, Jul 05 2017

Keywords

• sign

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).

Crossrefs

Cf. A139398, A133476, A139714, A139748, A139761.

Cf. A289306, A289321, A289387, A289389.

Programs

Mathematica

Table[Sum[(-1)^k*Binomial[n, 5 k + 3], {k, 0, n}], {n, 0, 36}] (* or *)
CoefficientList[Series[((-1 + x) x^3)/((-1 + x)^5 - x^5), {x, 0, 36}], x] (* Michael De Vlieger, Jul 10 2017 *)

PARI

a(n) = sum(k=0, (n-3)\5, (-1)^k*binomial(n, 5*k+3)); \\ Michel Marcus, Jul 05 2017

Formula

G.f.: ((-1+x)*x^3)/((-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-6)/10) + (phi-1)^n*cos (3* Pi*(n-6)/10)), where phi is the golden ratio.

a(n+m) = a(n)*K_1(m) + K_3(n)*K_2(m) + K_2(n)*K_3(m) + K_1(n)*a(m) - K_5(n)*K_5(m), where K_1 is A289306, K_2 is A289321, K_3 is A289387, K_5 is A289389.

a(n) = 0 if and only if n=0, n=2 or n==1 (mod 10). - Vladimir Shevelev, Jul 15 2017

Extensions

More terms from Peter J. C. Moses, Jul 05 2017