cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A347107 a(n) = Sum_{1 <= i < j <= n} j^3*i^3.

Original entry on oeis.org

0, 0, 8, 251, 2555, 15055, 63655, 214918, 616326, 1561110, 3586110, 7612385, 15139553, 28506101, 51229165, 88438540, 147420940, 238291788, 374813076, 575377095, 864177095, 1272587195, 1840775123, 2619572626, 3672629650, 5078879650, 6935344650, 9360309933
Offset: 0

Views

Author

Roudy El Haddad, Jan 27 2022

Keywords

Comments

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

Examples

			For n=3, a(3) = (2*1)^3+(3*1)^3+(3*2)^3 = 251.

Links

Crossrefs

Cf. A000537, A000578.
Cf. A346642 (for nondistinct cubes).
Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares).

Programs

  • Mathematica
    CoefficientList[Series[-(x^5 + 64 x^4 + 424 x^3 + 584 x^2 + 179 x + 8) x^2/(x - 1)^9, {x, 0, 27}], x] (* Michael De Vlieger, Feb 04 2022 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,8,251,2555,15055,63655,214918,616326},30] (* Harvey P. Dale, Jul 07 2025 *)
  • PARI
    a(n) = sum(i=2, n, sum(j=1, i-1, i^3*j^3));
  • PARI
    {a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672};
  • Python
    def A347107(n): return n*(n**2*(n*(n*(n*(n*(21*n + 36) - 42) - 84) + 21) + 56) - 8)//672 # Chai Wah Wu, Feb 17 2022

Formula

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^3*i^3.
a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672 (from the generalized form of Faulhaber's formula).
From Alois P. Heinz, Jan 27 2022: (Start)
a(n) = Sum_{i=1..n} A000578(i)*A000537(i-1) = Sum_{i=1..n} i^3*(i*(i-1)/2)^2.
G.f.: -(x^5+64*x^4+424*x^3+584*x^2+179*x+8)*x^2/(x-1)^9. (End)

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

Views

Author

Roudy El Haddad, Apr 13 2022

Keywords

Comments

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}.

References

  • 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.

Links

Crossrefs

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).

Programs

  • PARI
    {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};
  • Python
    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

Formula

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.

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

Views

Author

Roudy El Haddad, Feb 18 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A000217 (for power 0), A001296 (for power 1), A060493 (for squares), A346642 (for cubes).
Cf. A000583 (fourth powers), A000538 (sum of fourth powers).

Programs

  • PARI
    {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};
  • PARI
    a(n) = sum(j=1, n, sum(i=1, j, i^4*j^4));

Formula

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

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

Views

Author

Roudy El Haddad, Feb 18 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A001296 (for power 1), A060493 (for squares), A346642 (for cubes), A351766 (for fourth powers).
Cf. A000584 (fifth powers), A000539 (sum of fifth powers).

Programs

  • Maple
    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
  • PARI
    {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};
  • PARI
    a(n) = sum(j=1, n, sum(i=1, j, i^5*j^5));

Formula

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
Showing 1-4 of 4 results.

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