A294272 Sum of the fifth powers of the parts in the partitions of n into two parts.

0, 2, 33, 308, 1300, 4668, 12201, 30032, 61776, 123950, 220825, 389652, 630708, 1018808, 1539825, 2331968, 3347776, 4826682, 6657201, 9233300, 12333300, 16578452, 21571033, 28256208, 35970000, 46106918, 57617001, 72503732, 89176276, 110446800, 133987425

Offset

1,2

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)} i^5 + (n-i)^5.

From Colin Barker, Nov 20 2017: (Start)

G.f.: x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6).

a(n) = (1/192)*(n^2*(-16 + 80*n^2 + 3*(-31 + (-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]}], {n, 50}]
Table[Total[Flatten[IntegerPartitions[n, {2}]]^5], {n, 35}] (* or *) LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {0, 2, 33, 308, 1300, 4668, 12201, 30032, 61776, 123950, 220825, 389652, 630708}, 40] (* Harvey P. Dale, Jun 07 2025 *)

Prog

(PARI) concat(0, Vec(x^2*(2 + 31*x + 263*x^2 + 806*x^3 + 1748*x^4 + 2046*x^5 + 1708*x^6 + 806*x^7 + 238*x^8 + 31*x^9 + x^10) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ Colin Barker, Nov 20 2017
(Magma) [(n^2*(-16 + 80*n^2 + 3*(-31 + (-1)^n)*n^3 + 32*n^4))/192 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Crossrefs

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), this sequence (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

Sequence in context: A093992 A382997 A361887 * A231595 A100023 A336969

Adjacent sequences: A294269 A294270 A294271 * A294273 A294274 A294275

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 26 2017

Status

approved