A294304 Sum of the ninth powers of the parts of the partitions of n into two distinct parts.
0, 0, 513, 19684, 282340, 2215782, 12313161, 52404624, 186884496, 572351860, 1574304985, 3922174980, 9092033028, 19656178794, 40357579185, 78666720832, 147520415296, 265720871304, 464467582161, 786155279940, 1299155279940, 2091077378830, 3300704544313
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 (1,10,-10,-45,45,120,-120,-210,210,252,-252,-210,210,120,-120,-45,45,10,-10,-1,1).
Programs
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Mathematica
Table[Sum[i^9 + (n - i)^9, {i, Floor[(n-1)/2]}], {n, 30}] -
PARI
a(n) = sum(i=1, (n-1)\2, i^9 + (n-i)^9); \\ Michel Marcus, Nov 08 2017
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PARI
concat(vector(2), Vec(x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ Colin Barker, Nov 20 2017
Formula
a(n) = Sum_{i=1..floor((n-1)/2)} i^9 + (n-i)^9.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.
(End)