A001298 -id:A001298 - cp's OEIS frontend

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

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).

A151510 The triangle in A151338 read by rows upwards.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 1, 0, 1, 10, 25, 15, 1, 0, 1, 15, 65, 90, 31, 0, 0, 1, 21, 140, 350, 301, 56, 0, 0, 1, 28, 266, 1050, 1701, 938, 91, 0, 0, 1, 36, 462, 2646, 6951, 7686, 2737, 126, 0, 0, 1, 45, 750, 5880, 22827, 42315, 32725, 7455, 126, 0, 0

Offset: 0

N. J. A. Sloane, May 14 2009

Columns 1-5 are A000012, A000217, A001296, A001297, and A001298. - Nathaniel Johnston, Apr 30 2011

			Triangle begins:
1
1 0
1 1  0
1 3  1   0
1 6  7   1    0
1 10 25  15   1    0
1 15 65  90   31   0   0
1 21 140 350  301  56  0  0
1 28 266 1050 1701 938 91 0 0
...

Extended by Nathaniel Johnston, Apr 30 2011

cp's OEIS Frontend

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

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A151510 The triangle in A151338 read by rows upwards.

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Author

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A151510 The triangle in A151338 read by rows upwards.

Views

Author

Keywords

Comments

Examples

Extensions

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