Roudy El Haddad - cp's OEIS frontend

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

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.

A354021 a(n) = Sum_{1 <= i < j < k < m <= n} (mkj*i)^2.

Original entry on oeis.org

0, 0, 0, 0, 576, 21076, 296296, 2475473, 14739153, 68943381, 268880381, 909450751, 2742417535, 7522650135, 19058554515, 45123156390, 100771975590, 213877057086, 434042943246, 846542846578, 1593528150578

Offset: 0

Roudy El Haddad, May 14 2022

a(n) is the sum of all products of four distinct squares of positive integers up to n, i.e., the sum of all products of four distinct elements from the set of squares {1^2, ..., n^2}.

Winston de Greef, Table of n, a(n) for n = 0..10000
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. A353021 (for nondistinct squares).
Cf. A000290 (squares), A000330 (sum of squares), A000596 (for two squares),  A000597 (for three squares).
Cf. A001298 (for power 1).

{a(n) = n*(n + 1)*(n - 1)*(n - 2)*(n - 3)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(2*n - 5)*(5*n + 7)*(35*n^2 + 98*n + 72)/5443200};

a(n) = Sum_{m=4..n} Sum_{k=3..m-1} Sum_{j=2..k-1} Sum_{i=1..j-1} (m*k*j*i)^2.
a(n) = n*(n+1)*(n-1)*(n-2)*(n-3)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(2*n - 5)*(5*n + 7)*(35*n^2 + 98*n + 72)/5443200.
a(n) = binomial(2*n+2,9)*(5*n + 7)*(35*n^2 + 98*n + 72)/(5!*4).

A352979 a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (kji)^3.

Original entry on oeis.org

0, 1, 585, 28800, 505280, 4951530, 33209946, 170320080, 714724560, 2566030995, 8130545995, 23253835176, 61054704360, 149085989780, 342048076020, 743408003520, 1540821690816, 3062326169925, 5862986735085, 10855192630480, 19500255870480

Offset: 0

Roudy El Haddad, Apr 13 2022

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

El Haddad, R. (2022). A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). [Typo corrected by _Georg Fischer_, Sep 30 2022]

Cf. A352980 (for distinct cubes).
Cf. A001297 (for power 1), A351105 (for squares).
Cf. A000578 (cubes), A000537 (sum of first n cubes), A346642 (order 2).

{a(n) = n^2 * (n + 1)^2 * (n + 2) * (n + 3) * (35*n^6 + 205*n^5 + 263*n^4 - 221*n^3 - 214*n^2 + 324*n - 112)/13440};

def A352979(n): return n**2*(n*(n*(n*(n*(n*(n*(n*(n*(n*(35*n + 450) + 2293) + 5700) + 6405) + 770) - 3661) - 240) + 2320) + 40) - 672)//13440 # Chai Wah Wu, May 14 2022

a(n) = n^2 * (n + 1)^2 * (n + 2) * (n + 3) * (35*n^6 + 205*n^5 + 263*n^4 - 221*n^3 - 214*n^2 + 324*n - 112)/13440.

a(n) = binomial(n+3,4)*binomial(n+1,2)*(35*n^6 + 205*n^5 + 263*n^4 - 221*n^3 - 214*n^2 + 324*n - 112)/280.

A353021 a(n) = Sum_{l=1..n} Sum_{k=1..l} Sum_{j=1..k} Sum_{i=1..j} (lkj*i)^2.

Original entry on oeis.org

0, 1, 341, 13013, 196053, 1733303, 10787231, 52253971, 209609235, 725520510, 2230238010, 6217887390, 15973440990, 38276304066, 86383520146, 185042663146, 378620563178, 743881306623, 1409531082531, 2585397711611, 4605062303611

Offset: 0

Roudy El Haddad, Apr 17 2022

a(n) is the sum of all products of four squares of positive integers up to n, i.e., the sum of all products of four elements from the set of squares {1^2, ..., n^2}.

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. See Theorem 4.8 for m = 4 and p = 2.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A354021 (for distinct squares).
Cf. A000290 (squares), A000330 (sum of squares), A060493 (for two squares), A351105 (for three squares).
Cf. A000915 (for power 1).

{a(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(2*n + 1)*(2*n + 3)*(2*n + 5)*(2*n + 7)*(5*n - 2)*(35*n^2 - 28*n + 9)/5443200};

def A353021(n): return n*(n*(n*(n*(n*(n*(n*(n*(8*n*(n*(70*n*(5*n + 84) + 40417) + 144720) + 2238855) + 2050020) + 207158) - 810600) - 58505) + 322740) + 7956) - 45360)//5443200 # Chai Wah Wu, May 14 2022

a(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(2*n + 1)*(2*n + 3)*(2*n + 5)*(2*n + 7)*(5*n - 2)*(35*n^2 - 28*n + 9)/5443200.

a(n) = binomial(2*n+8,9)*(5*n - 2)*(35*n^2 - 28*n + 9)/(5!*4).

A351864 Numerator of zeta({6}_n)/Pi^(6n).

Original entry on oeis.org

1, 1, 4, 2, 4, 1, 4, 4, 4, 4, 16, 2, 4, 2, 8, 8, 4, 4, 16, 8, 16, 1, 4, 4, 4, 4, 16, 4, 8, 4, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 2, 4, 2, 8, 8, 4, 4, 16, 8, 16, 2, 8, 8, 8, 8, 32, 8, 16, 8, 32, 32, 4, 4, 16, 8, 16, 4, 16

Offset: 0

Roudy El Haddad, Feb 22 2022

({6}_n) is standard notation for multiple zeta values. It represents (6, ..., 6) where the multiplicity of 6 is n.

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, arXiv:hep-th/9611004, 1996.
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.

Cf. A351806 (denominators).

Cf. A000120, A240883, A046988, A013664, A103345.

a[n_] := Numerator[6*2^(6*n)/(6*n + 3)!]; Array[a, 71, 0]

PARI
```
a(n) = 1 << (hammingweight(3*n+1) - 1);
```

a(n) = numerator(6*2^(6*n)/(6*n + 3)!).
a(n) = 2^(A000120(3*n + 1) - 1).
a(n) = 2^A240883(n).

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

A351806 Denominator of zeta({6}_n)/Pi^(6*n).

Original entry on oeis.org

1, 945, 212837625, 64965492466875, 432684797065192546875, 1347828286825972065254765625, 197885500589205605585596463448046875, 18132629348577543860598956218936672646484375, 3673787208165374996876652878250276546299488037109375

Offset: 0

Roudy El Haddad, Feb 19 2022

({6}_n) is standard notation for multiple zeta values. It represents (6, ..., 6) where the multiplicity of 6 is n.

J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, arXiv:hep-th/9611004, 1996.
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.

Cf. A351864 (numerators).
Cf. A002432 (denominators of zeta(2*n)/Pi^(2*n)).
Cf. A013664 (zeta(6)).
Cf. A103345.

a[n_] := Denominator[6*2^(6*n)/(6*n + 3)!]; Array[a, 9, 0] (* Amiram Eldar, Feb 19 2022 *)

a(n) = denominator(6*2^(6*n)/(6*n + 3)!); \\ Michel Marcus, Feb 22 2022

a(n) = denominator(6*2^(6*n)/(6*n + 3)!).

A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.

Original entry on oeis.org

0, 1, 1057, 68125, 1399325, 15227450, 110102426, 597639882, 2621915850, 9756511275, 31839011275, 93340522951, 250280856007, 622316813300, 1450471654100, 3196426654100, 6706824221076, 13476181309557, 26055415288725, 48670370285425, 88136930285425, 155187254126926

Offset: 0

Roudy El Haddad, Feb 18 2022

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

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A001296 (for power 1), A060493 (for squares), A346642 (for cubes), A351766 (for fourth powers).

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

seq(n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168,
n=0..30);# Robert Israel, Feb 18 2022

{a(n) = n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168};

a(n) = sum(j=1, n, sum(i=1, j, i^5*j^5));

a(n) = n*(n+1)*(n+2)*(44*n^9 + 276*n^8 + 492*n^7 - 48*n^6 - 609*n^5 + 207*n^4 + 487*n^3 - 291*n^2 - 90*n + 60)/3168.

G.f.: x*(1 + 1044*x + 54462*x^2 + 595860*x^3 + 2048388*x^4 + 2563644*x^5 + 1193226*x^6 + 188508*x^7 + 7635*x^8 + 32*x^9)/(1-x)^13. - Robert Israel, Feb 18 2022

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

A351766 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^4.

Original entry on oeis.org

0, 1, 273, 8211, 98835, 710710, 3659110, 14886186, 50816298, 151416111, 404746111, 990005445, 2248888005, 4798557036, 9703780828, 18730825828, 34711648356, 62053834605, 107439683325, 180766879111, 296393439111, 474761104818, 744484165986, 1145004918190, 1729932641710, 2571200219835

Offset: 0

Roudy El Haddad, Feb 18 2022

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

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000217 (for power 0), A001296 (for power 1), A060493 (for squares), A346642 (for cubes).

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

{a(n) = n*(n+1)*(n+2)*(2*n+1)*(2*n+3)*(9*n^5+25*n^4-5*n^3-25*n^2+21*n-5)/1800};

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

a(n) = n*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3)*(9*n^5 + 25*n^4 - 5*n^3 - 25*n^2 + 21*n - 5)/1800.
a(n) = binomial(2*n+4,5) * (9*n^5 + 25*n^4 - 5*n^3 - 25*n^2 + 21*n - 5)/5!.
G.f.: x*(16*x^7 + 1217*x^6 + 12038*x^5 + 30415*x^4 + 23364*x^3 + 5263*x^2 + 262*x + 1)/(1 - x)^11. - Alois P. Heinz, Feb 18 2022

cp's OEIS Frontend

User: Roudy El Haddad

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

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A354021 a(n) = Sum_{1 <= i < j < k < m <= n} (m*k*j*i)^2.

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A352979 a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (k*j*i)^3.

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A353021 a(n) = Sum_{l=1..n} Sum_{k=1..l} Sum_{j=1..k} Sum_{i=1..j} (l*k*j*i)^2.

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A351864 Numerator of zeta({6}_n)/Pi^(6n).

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

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A351806 Denominator of zeta({6}_n)/Pi^(6*n).

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A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.

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

A354021 a(n) = Sum_{1 <= i < j < k < m <= n} (mkj*i)^2.

A352979 a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (kji)^3.

A353021 a(n) = Sum_{l=1..n} Sum_{k=1..l} Sum_{j=1..k} Sum_{i=1..j} (lkj*i)^2.