1,3

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,8,-8,-28,28,56,-56,-70,70,56,-56,-28,28,8,-8,-1,1).

a(n) = Sum_{i=1..floor((n-1)/2)} i^7 + (n-i)^7.

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8).

a(n) = (1/768)*(n^2*(64 - 224*n^2 + 448*n^4 - 3*(129 + (-1)^n)*n^5 + 96*n^6)).

a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - 28*a(n-4) + 28*a(n-5) + 56*a(n-6) - 56*a(n-7) - 70*a(n-8) + 70*a(n-9) + 56*a(n-10) - 56*a(n-11) - 28*a(n-12) + 28*a(n-13) + 8*a(n-14) - 8*a(n-15) - a(n-16) + a(n-17) for n>17.

(End)

Table[Sum[i^7 + (n - i)^7, {i, Floor[(n-1)/2]}], {n, 40}]

CoefficientList[Series[x^3(129+2059x+15480x^2+59466x^3+153639x^4+257307x^5+ 311664x^6+ 258532x^7+153639x^8+60537x^9+15480x^10+2178x^11+129x^12+x^13)/ ((1-x)^9 (1+x)^8), {x, 0, 60}], x] (* or *) LinearRecurrence[{1, 8, -8, -28, 28, 56, -56, -70, 70, 56, -56, -28, 28, 8, -8, -1, 1}, {0, 0, 0, 129, 2188, 18700, 94638, 376761, 1183920, 3297456, 8002300, 18080425, 37287660, 73399404, 135324378, 241561425, 410323648}, 60] (* Harvey P. Dale, Aug 05 2021 *)

(PARI) a(n) = sum(i=1, (n-1)\2, i^7 + (n-i)^7); \\ Michel Marcus, Nov 08 2017

(PARI) concat(vector(2), Vec(x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ Colin Barker, Nov 20 2017

Cf. A294286, A294287, A294288, A294300, A294301.

Sequence in context: A351302 A017677 A013955 * A343509 A221969 A036085

Adjacent sequences: A294299 A294300 A294301 * A294303 A294304 A294305

nonn,easy

Wesley Ivan Hurt, Oct 27 2017

approved