OFFSET
0,3
COMMENTS
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}.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
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 Number Theory Disc. Math. 28(2) (2022), 167-199.
Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
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).
a(n) = (1/(9!*2))*((2n+6)!/(2n-1)!)*(35*n^2-21*n+4).
a(n) = binomial(2n+6,7)*(35*n^2-21*n+4)/144.
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
a(n) = 2/(2*n)! * Sum_{j = 1..n} (-1)^(n+j) * binomial(2*n, n-j) * j^(2*n+6). - Peter Bala, Mar 31 2025
MATHEMATICA
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 *)
(* Alternative: *)
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 *)
PROG
(PARI) {a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+1)*(2*n+3)*(2*n+5)*(35*n^2-21*n+4)/45360};
(PARI) a(n) = sum(i=1, n, sum(j=1, i, sum(k=1, j, i^2*j^2*k^2)));
(Python)
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roudy El Haddad, Jan 31 2022
STATUS
approved