A294300 Sum of the fifth powers of the parts in the partitions of n into two distinct parts.

0, 0, 33, 244, 1300, 4182, 12201, 27984, 61776, 117700, 220825, 374100, 630708, 985194, 1539825, 2266432, 3347776, 4708584, 6657201, 9033300, 12333300, 16256350, 21571033, 27758544, 35970000, 45364332, 57617001, 71428084, 89176276, 108928050, 133987425

Offset

1,3

Links

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

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Formula

a(n) = Sum_{i=1..floor(n/2)-((n+1) mod 2)} i^5 + (n-i)^5.

G.f.: -x^3*(33 +211*x +858*x^2 +1616*x^3 +2178*x^4 +1656*x^5 +858*x^6 +236*x^7 +33*x^8 +x^9) /(1+x)^6 /(x-1)^7. - R. J. Mathar, Nov 07 2017

From Colin Barker, Nov 21 2017: (Start)

a(n) = (1/192)*(n^2*(-16 + 80*n^2 - 3*(33 + (-1)^n)*n^3 + 32*n^4)).

a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>13.

(End)

Mathematica

Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2] - Mod[n + 1, 2]}], {n, 40}]

Prog

(PARI) concat(vector(2), Vec(x^3*(33 + 211*x + 858*x^2 + 1616*x^3 + 2178*x^4 + 1656*x^5 + 858*x^6 + 236*x^7 + 33*x^8 + x^9) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ Colin Barker, Nov 21 2017
(PARI) a(n) = sum(i=1, (n-1)\2, i^5 + (n-i)^5); \\ Michel Marcus, Nov 22 2017

Crossrefs

Cf. A294286, A294287, A294288.

Sequence in context: A351300 A017673 A001160 * A271208 A343499 A318744

Adjacent sequences: A294297 A294298 A294299 * A294301 A294302 A294303

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 27 2017

Status

approved