A347107 -id:A347107 - cp's OEIS frontend

A346642 a(n) = Sum_{j=1..n} Sum_{i=1..j} j^3*i^3.

0, 1, 73, 1045, 7445, 35570, 130826, 399738, 1063290, 2539515, 5564515, 11362351, 21875503, 40068860, 70321460, 118921460, 194681076, 309689493, 480223005, 727832905, 1080632905, 1574809126, 2256376958, 3183210350, 4427370350, 6077760975, 8243141751

Offset: 0

Roudy El Haddad, Jan 24 2022

a(n) is the sum of all products of two cubes of positive integers up to n, i.e., the sum of all products of two elements from the set of cubes {1^3, ..., n^3}.

			For n=3,
a(3) = (1*1)^3+(2*1)^3+(2*2)^3+(3*1)^3+(3*2)^3+(3*3)^3 = 1045,
a(3) = 1^3*(1^3)+2^3*(1^3+2^3)+3^3*(1^3+2^3+3^3) = 1045.

Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000537 (sum of first n cubes), A347107 (for distinct cubes).

Cf. A001296 (for power 1), A060493 (for squares).

CoefficientList[Series[-(8 x^5 + 179 x^4 + 584 x^3 + 424 x^2 + 64 x + 1) x/(x - 1)^9, {x, 0, 26}], x] (* Michael De Vlieger, Feb 04 2022 *)

{a(n) = n*(n+1)*(n+2)*(21n^5+69n^4+45n^3-21n^2-6n+4)/672};

a(n) = sum(i=1, n, sum(j=1, i, i^3*j^3)); \\ Michel Marcus, Jan 27 2022

def A346642(n): return n*(n**2*(n*(n*(n*(n*(21*n + 132) + 294) + 252) + 21) - 56) + 8)//672 # Chai Wah Wu, Feb 17 2022

a(n) = n*(n+1)*(n+2)*(21*n^5+69*n^4+45*n^3-21*n^2-6*n+4)/672 (from the recurrent form of Faulhaber's formula).

G.f.: -(8*x^5+179*x^4+584*x^3+424*x^2+64*x+1)*x/(x-1)^9. - Alois P. Heinz, Jan 27 2022

A352980 a(n) = Sum_{1 <= i < j < k <= n} (kji)^3.

Original entry on oeis.org

0, 0, 0, 216, 16280, 335655, 3587535, 25421200, 135459216, 584760870, 2145870870, 6918983280, 20073184560, 53334782501, 131555523645, 304453955520, 666698215360, 1390977293580, 2780695001196, 5351537889480, 9954554649480, 17957698726275

Offset: 0

Roudy El Haddad, Apr 13 2022

a(n) is the sum of all products of three distinct cubes of positive integers up to n, i.e., the sum of all products of three distinct elements from the set of cubes {1^3, ..., n^3}.

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2) 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233. See Theorem 5.1 for m = 3 and p = 3.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). [Typo corrected by _Georg Fischer_, Sep 30 2022]

Cf. A352979 (for nondistinct cubes).
Cf. A001303 (for power 1), A000597 (for squares).
Cf. A000578 (cubes), A000537 (sum of first n cubes), A347107 (order 2).

{a(n) = n^2 * (n + 1)^2 * (n - 1) * (n - 2) * (35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/13440};

def A352980(n): return n**2*(n*(n*(n*(n*(n*(n*(n*(n*(n*(35*n - 30) - 347) + 180) + 1365) - 350) - 2541) + 240) + 2160) - 40) - 672)//13440 # Chai Wah Wu, May 15 2022

a(n) = Sum_{k=3..n} Sum_{j=2..k-1} Sum_{i=1..j-1} k^3*j^3*i^3.
a(n) = n^2 * (n + 1)^2 * (n - 1) * (n - 2) * (35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/13440.
a(n) = binomial(n+1,4)*binomial(n+1,2)*(35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/280.

A351760 a(n) = Sum_{1 <= i < j <= n} (i*j)^4.

Original entry on oeis.org

0, 0, 16, 1393, 26481, 247731, 1516515, 6978790, 26131686, 83684778, 237014778, 607915231, 1436816095, 3170754405, 6600189141, 13064343516, 24750198748, 45116627556, 79482515700, 135826148445, 225852708445, 366397514791, 581244702423, 903454469346, 1378306878690, 2066986566190

Offset: 0

Roudy El Haddad, Feb 18 2022

a(n) is the sum of all products of two distinct elements from the set {1^4, ..., n^4}.

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares), A347107 (for cubes).

Cf. A000583 (fourth powers), A000538 (sum of fourth powers).

{a(n) = n*(n-1)*(n+1)*(2*n-1)*(2*n+1)*(9*n^5+20*n^4-15*n^3-50*n^2+n+30)/1800};

a(n) = sum(j=2, n, sum(i=1, j-1, i^4*j^4));

def A351760(n): return n*(n*(n*(n*(n*(n*(n*(n*(n*(9*n+20<<2)-105)-300)+88)+390)-20)-200)+1)+30)//1800 # Chai Wah Wu, Oct 03 2024

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^4*i^4.
a(n) = n*(n - 1)*(n + 1)*(2*n - 1)*(2*n + 1)*(9*n^5 + 20*n^4 - 15*n^3 - 50*n^2 + n + 30)/1800.
a(n) = binomial(2*n+2, 5)*(9*n^5 + 20*n^4 - 15*n^3 - 50*n^2 + n + 30)/5!.
G.f.: x^2*(16 + 1217*x + 12038*x^2 + 30415*x^3 + 23364*x^4 + 5263*x^5 + 262*x^6 + x^7)/(1 - x)^11. - Stefano Spezia, Feb 18 2022

A351805 a(n) = Sum_{1 <= i < j <= n} j^5*i^5.

Original entry on oeis.org

0, 0, 32, 8051, 290675, 4353175, 38761975, 243824182, 1194358326, 4842169350, 16924669350, 52488756425, 147511725257, 381689190701, 920589376525, 2089893985900, 4500779925100, 9254143113132, 18262909865676, 34746798604575, 63973358604575, 114343801467875

Offset: 0

Roudy El Haddad, Feb 19 2022

a(n) is the sum of all products of two distinct elements from the set {1^5, ..., n^5}.

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares), A347107 (for cubes), (for fourth powers).

Cf. A000584 (fifth powers), A000539 (sum of fifth powers).

{a(n) = n*(n-1)*(n+1)*(44*n^9+120*n^8-132*n^7-540*n^6+99*n^5+912*n^4-11*n^3-672*n^2+120)/3168};

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^5*i^5.
a(n) = n*(n - 1)*(n + 1)*(44*n^9 + 120*n^8 - 132*n^7 - 540*n^6 + 99*n^5 + 912*n^4 - 11*n^3 - 672*n^2 + 120)/3168.
G.f.: -x^2*(x^9 +1044*x^8 +54462*x^7 +595860*x^6 +2048388*x^5 +2563644*x^4 +1193226*x^3 +188508*x^2 +7635*x +32)/(x-1)^13. - Alois P. Heinz, Feb 19 2022

cp's OEIS Frontend

A346642 a(n) = Sum_{j=1..n} Sum_{i=1..j} j^3*i^3.

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A352980 a(n) = Sum_{1 <= i < j < k <= n} (k*j*i)^3.

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A351760 a(n) = Sum_{1 <= i < j <= n} (i*j)^4.

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A351805 a(n) = Sum_{1 <= i < j <= n} j^5*i^5.

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A352980 a(n) = Sum_{1 <= i < j < k <= n} (kji)^3.