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A346642 a(n) = Sum_{j=1..n} Sum_{i=1..j} j^3*i^3. 5
0, 1, 73, 1045, 7445, 35570, 130826, 399738, 1063290, 2539515, 5564515, 11362351, 21875503, 40068860, 70321460, 118921460, 194681076, 309689493, 480223005, 727832905, 1080632905, 1574809126, 2256376958, 3183210350, 4427370350, 6077760975, 8243141751 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the sum of all products of two cubes of positive integers up to n, i.e., the sum of all products of two elements from the set of cubes {1^3, ..., n^3}.

LINKS

Table of n, a(n) for n=0..26.

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 (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

a(n) = n*(n+1)*(n+2)*(21*n^5+69*n^4+45*n^3-21*n^2-6*n+4)/672 (from the recurrent form of Faulhaber's formula).

G.f.: -(8*x^5+179*x^4+584*x^3+424*x^2+64*x+1)*x/(x-1)^9. - Alois P. Heinz, Jan 27 2022

EXAMPLE

For n=3,

a(3) = (1*1)^3+(2*1)^3+(2*2)^3+(3*1)^3+(3*2)^3+(3*3)^3 = 1045,

a(3) = 1^3*(1^3)+2^3*(1^3+2^3)+3^3*(1^3+2^3+3^3) = 1045.

MATHEMATICA

CoefficientList[Series[-(8 x^5 + 179 x^4 + 584 x^3 + 424 x^2 + 64 x + 1) x/(x - 1)^9, {x, 0, 26}], x] (* Michael De Vlieger, Feb 04 2022 *)

PROG

(PARI) {a(n) = n*(n+1)*(n+2)*(21n^5+69n^4+45n^3-21n^2-6n+4)/672};

(PARI) a(n) = sum(i=1, n, sum(j=1, i, i^3*j^3)); \\ Michel Marcus, Jan 27 2022

(Python)

def A346642(n): return n*(n**2*(n*(n*(n*(n*(21*n + 132) + 294) + 252) + 21) - 56) + 8)//672 # Chai Wah Wu, Feb 17 2022

CROSSREFS

Cf. A000537 (sum of first n cubes), A347107 (for distinct cubes).

Cf. A001296 (for power 1), A060493 (for squares).

Sequence in context: A320205 A305549 A320214 * A008400 A090685 A232297

Adjacent sequences: A346639 A346640 A346641 * A346643 A346644 A346645

KEYWORD

nonn,easy

AUTHOR

Roudy El Haddad, Jan 24 2022

STATUS

approved

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Last modified December 4 21:40 EST 2022. Contains 358570 sequences. (Running on oeis4.)