A351105 a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (i*j*k)^2.

0, 1, 85, 1408, 11440, 61490, 251498, 846260, 2458676, 6369275, 15047175, 32955780, 67746900, 131969604, 245444980, 438485080, 756163672, 1263878005, 2054474617, 3257248280, 5049161480, 7668672374, 11432601950, 16756516140, 24179145900, 34391417775

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 on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.

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

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 *)
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)*(2n+1)*(2n+3)*(2n+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

A diagonal of A036969.

Cf. A000290 (squares), A000330 (sum of squares), A060493 (for two squares).

Cf. A001297 (for power 1).

Sequence in context: A189439 A166917 A206377 * A008360 A212474 A020310

Adjacent sequences: A351102 A351103 A351104 * A351106 A351107 A351108

Keyword

nonn,easy

Author

Roudy El Haddad, Jan 31 2022

Status

approved