A361610 - OEIS

%I #17 Aug 29 2024 17:17:47

%S 1,70,1175,13500,128125,1081250,8421875,61875000,434765625,2949218750,

%T 19443359375,125195312500,790283203125,4904785156250,29998779296875,

%U 181152343750000,1081695556640625,6394958496093750,37471771240234375,217819213867187500,1257038116455078125

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

%C 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).

%H Winston de Greef, <a href="/A361610/b361610.txt">Table of n, a(n) for n = 0..1408</a>

%H Project Euler, <a href="https://projecteuler.net/problem=831">Problem 831. Triple Product</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (20,-150,500,-625).

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

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

%F 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.

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

%o (Python)

%o def A361610(n): return 5**n*(n*(n*(4*n + 18) + 17) + 3)//3 # _Chai Wah Wu_, Mar 17 2023

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

%K nonn,easy

%O 0,2

%A _R. J. Mathar_, Mar 17 2023