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A294301
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Sum of the sixth powers of the parts in the partitions of n into two distinct parts.
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5
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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
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history;
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OFFSET
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1,3
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LINKS
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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).
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FORMULA
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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)
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MATHEMATICA
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Table[Sum[i^6 + (n - i)^6, {i, Floor[(n-1)/2]}], {n, 40}]
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i^6 + (n-i)^6); \\ Michel Marcus, Nov 08 2017
(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
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CROSSREFS
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Cf. A294286, A294287, A294288, A294300.
Sequence in context: A351301 A017675 A013954 * A343508 A116277 A220389
Adjacent sequences: A294298 A294299 A294300 * A294302 A294303 A294304
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KEYWORD
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nonn,easy
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AUTHOR
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Wesley Ivan Hurt, Oct 27 2017
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STATUS
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approved
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