A361610 a(n) = 5^n*(n+1)*(4*n^2+14*n+3)/3.

1, 70, 1175, 13500, 128125, 1081250, 8421875, 61875000, 434765625, 2949218750, 19443359375, 125195312500, 790283203125, 4904785156250, 29998779296875, 181152343750000, 1081695556640625, 6394958496093750, 37471771240234375, 217819213867187500, 1257038116455078125

Offset: 0

Author

R. J. Mathar, Mar 17 2023

Keywords

• nonn
• easy

Comments

The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) * binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=3) = a(n).

Links

• Winston de Greef, Table of n, a(n) for n = 0..1408
• Project Euler, Problem 831. Triple Product
• Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Crossrefs

Cf. A027471 (k=1), A361609 (k=2), A361608 (k=5).

Programs

Mathematica

LinearRecurrence[{20,-150,500,-625},{1,70,1175,13500},30] (* Harvey P. Dale, Aug 29 2024 *)

Python

def A361610(n): return 5**n*(n*(n*(4*n + 18) + 17) + 3)//3 # Chai Wah Wu, Mar 17 2023

Formula

G.f.: (1 + 50*x - 75*x^2) / (5*x - 1)^4.

a(n) = 20*a(n-1) -150*a(n-2) +500*a(n-3) -625*a(n-4).

D-finite with recurrence n*(4*n^2+6*n-7)*a(n) -5*(n+1)*(4*n^2+14*n+3)*a(n-1)=0.