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
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).
Programs
-
Mathematica
Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2] - Mod[n + 1, 2]}], {n, 40}] Table[Total[Flatten[Select[IntegerPartitions[n,{2}],#[[1]]!=#[[2]]&]]^5],{n,40}] (* Harvey P. Dale, Sep 04 2024 *) -
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
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)