A294303 Sum of the eighth powers of the parts in the partitions of n into two distinct parts.

0, 0, 257, 6562, 72354, 456418, 2142595, 7841860, 24684612, 67340708, 167731333, 380410598, 812071910, 1622037830, 3103591687, 5649705096, 9961449608, 16894160328, 27957167625, 44840730666, 70540730666, 108149231146, 163239463563, 241120467148, 351625763020

Offset

1,3

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-36,36,84,-84,-126,126,126,-126,-84,84,36,-36,-9,9,1,-1).

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i^8 + (n-i)^8.

From David A. Corneth, Nov 05 2017: (Start)

For odd n, a(n) = n^9 / 9 - n^8/2 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30

For even n, a(n) = n^9 / 9 - 129*n^8/256 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30.

(End)

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9).

a(n) = a(n-1) + 9*a(n-2) - 9*a(n-3) - 36*a(n-4) + 36*a(n-5) + 84*a(n-6) - 84*a(n-7) - 126*a(n-8) + 126*a(n-9) + 126*a(n-10) - 126*a(n-11) - 84*a(n-12) + 84*a(n-13) + 36*a(n-14) - 36*a(n-15) - 9*a(n-16) + 9*a(n-17) + a(n-18) - a(n-19) for n>19.

(End)

Mathematica

Table[Sum[i^8 + (n - i)^8, {i, Floor[(n-1)/2]}], {n, 30}]

Prog

(PARI) first(n) = {my(res = vector(n, i, 1/9*i^9 - 1/2*i^8 + 2/3*i^7 - 7/15*i^5 + 2/9*i^3 - 1/30*i)); forstep(i = 2, #res, 2, res[i] -= i^8/256); res} \\ David A. Corneth, Nov 05 2017
(PARI) concat(vector(2), Vec(x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ Colin Barker, Nov 20 2017

Crossrefs

Cf. A294286, A294287, A294288, A294300, A294301, A294302.

Sequence in context: A351303 A017679 A013956 * A036086 A000542 A023877

Adjacent sequences: A294300 A294301 A294302 * A294304 A294305 A294306

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 27 2017

Status

approved