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A353021
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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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2
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0, 1, 341, 13013, 196053, 1733303, 10787231, 52253971, 209609235, 725520510, 2230238010, 6217887390, 15973440990, 38276304066, 86383520146, 185042663146, 378620563178, 743881306623, 1409531082531, 2585397711611, 4605062303611
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OFFSET
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0,3
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COMMENTS
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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}.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
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FORMULA
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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).
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PROG
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(PARI) {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};
(Python)
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
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CROSSREFS
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Cf. A354021 (for distinct squares).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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