A361610 - OEIS

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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

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OFFSET

0,2

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).

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.

MATHEMATICA

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

PROG

(Python)