A361609 a(n) = 4^n*(1 + (23/8)*n + (9/8)*n^2).

1, 20, 180, 1264, 7808, 44544, 240640, 1249280, 6291456, 30932992, 149159936, 707788800, 3313500160, 15334375424, 70262980608, 319169757184, 1438814044160, 6442450944000, 28673201668096, 126924873531392, 559101662724096, 2451910929940480, 10709243254538240, 46601700831657984

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=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).

Crossrefs

Cf. A027471 (k=1), A361610 (k=3), A361608 (k=5).

Programs

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 *)

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

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.