A294301 Sum of the sixth powers of the parts in the partitions of n into two distinct parts.
0, 0, 65, 730, 4890, 19786, 67171, 180724, 446964, 962780, 1978405, 3703310, 6735950, 11445110, 19092295, 30220776, 47260136, 70866264, 105409929, 151455810, 216455810, 300450370, 415601835, 560651740, 754740700, 994054516, 1307797101, 1687688054, 2177107894
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 (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).
Programs
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Mathematica
Table[Sum[i^6 + (n - i)^6, {i, Floor[(n-1)/2]}], {n, 40}] -
PARI
a(n) = sum(i=1, (n-1)\2, i^6 + (n-i)^6); \\ Michel Marcus, Nov 08 2017
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PARI
concat(vector(2), Vec(x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ Colin Barker, Nov 20 2017
Formula
a(n) = Sum_{i=1..floor((n-1)/2)} i^6 + (n-i)^6.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7).
a(n) = (n/42 - n^3/6 + n^5/2 - 1/128*(65 + (-1)^n)*n^6 + n^7/7).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 21*a(n-4) + 21*a(n-5) + 35*a(n-6) - 35*a(n-7) - 35*a(n-8) + 35*a(n-9) + 21*a(n-10) - 21*a(n-11) - 7*a(n-12) + 7*a(n-13) + a(n-14) - a(n-15) for n>15.
(End)