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%I #38 Jul 18 2022 16:29:29
%S 0,1,85,1408,11440,61490,251498,846260,2458676,6369275,15047175,
%T 32955780,67746900,131969604,245444980,438485080,756163672,1263878005,
%U 2054474617,3257248280,5049161480,7668672374,11432601950,16756516140,24179145900,34391417775
%N a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (i*j*k)^2.
%C a(n) is the sum of all products of three squares of positive integers up to n, i.e., the sum of all products of three elements from the set of squares {1^2, ..., n^2}.
%H Harvey P. Dale, <a href="/A351105/b351105.txt">Table of n, a(n) for n = 0..1000</a>
%H Roudy El Haddad, <a href="https://arxiv.org/abs/2101.09089">Recurrent Sums and Partition Identities</a>, arXiv:2101.09089 [math.NT], 2021.
%H Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.167-199">A generalization of multiple zeta value. Part 1: Recurrent sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F a(n) = n*(n+1)*(n+2)*(n+3)*(2n+1)*(2n+3)*(2n+5)*(35*n^2-21*n+4)/45360 (from the recurrent form of Faulhaber's formula).
%F a(n) = (1/(9!*2))*((2n+6)!/(2n-1)!)*(35*n^2-21*n+4).
%F a(n) = binomial(2n+6,7)*(35*n^2-21*n+4)/144.
%F G.f.: x*(36*x^5+460*x^4+1065*x^3+603*x^2+75*x+1)/(x-1)^10. - _Alois P. Heinz_, Jan 31 2022
%t CoefficientList[Series[x (36 x^5 + 460 x^4 + 1065 x^3 + 603 x^2 + 75 x + 1)/(x - 1)^10, {x, 0, 25}], x] (* _Michael De Vlieger_, Feb 04 2022 *)
%t LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,85,1408,11440,61490,251498,846260,2458676,6369275},30] (* _Harvey P. Dale_, Jul 18 2022 *)
%o (PARI) {a(n) = n*(n+1)*(n+2)*(n+3)*(2n+1)*(2n+3)*(2n+5)*(35*n^2-21*n+4)/45360};
%o (PARI) a(n) = sum(i=1, n, sum(j=1, i, sum(k=1, j, i^2*j^2*k^2)));
%o (Python)
%o def A351105(n): return n*(n*(n*(n*(n*(n*(n*(n*(280*n + 2772) + 10518) + 18711) + 14385) + 1323) - 2863) - 126) + 360)//45360 # _Chai Wah Wu_, Feb 17 2022
%Y A diagonal of A036969.
%Y Cf. A000290 (squares), A000330 (sum of squares), A060493 (for two squares).
%Y Cf. A001297 (for power 1).
%K nonn,easy
%O 0,3
%A _Roudy El Haddad_, Jan 31 2022
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