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=2) = a(n).
LINKS
Winston de Greef, Table of n, a(n) for n = 0..1640
Project Euler, Problem 831. Triple Product
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).
FORMULA
G.f.: ( -1-8*x+12*x^2 ) / (4*x-1)^3.
a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3).
D-finite with recurrence (-9*n^2-5*n+6)*a(n) +4*(9*n^2+23*n+8)*a(n-1)=0.
MATHEMATICA
LinearRecurrence[{12, -48, 64}, {1, 20, 180}, 25] (* or *)
A361609[n_] := 4^n (1 + 23/8 n + 9/8 n^2);
Array[A361609, 25, 0] (* Paolo Xausa, Jan 18 2024 *)
PROG
(Python)
def A361609(n): return (n*(9*n + 23) + 8)<<((n<<1)-3) if n > 1 else 19*n+1 # Chai Wah Wu, Mar 17 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Mar 17 2023
STATUS
approved