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

Author

Roudy El Haddad, Jan 31 2022

Keywords

• nonn
• easy

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).

Crossrefs

A diagonal of A036969.

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

Cf. A001297 (for power 1).

Programs

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 *)

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

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