0,2

5 divides a(n) except when a(n) == 1 (mod 5), which occurs only when the base 5 representation of n consists solely of digits '0' and '2' (conjecture). Positions n at which a(n) == 1 (mod 5) begin: [0, 2, 10,12, 50,52,60,62, 250,252,260,262,300,302,310,312, 1250,1252,1260,1262,1300,1302,1310,1312,1500,1502,1510,1512, 1550,1552,1560,1562, ...].

Vincenzo Librandi, Table of n, a(n) for n = 0..200

V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, Sep 20 2012

a(n) ~ 3^n*(1+(2/3)^(1/3))^(3*n+2)/(2^(4/3)*3^(1/6)*Pi*n)*(1+((2/3)^(1/3)+(3/2)^(1/3)-5)/(9*n)). - Vaclav Kotesovec, Sep 30 2012

G.f.: hypergeom([1/3, 2/3],[1],162*x^2/(1-5*x)^3)/(1-5*x). - Mark van Hoeij, Apr 11 2014

D-finite with recurrence 2*(n^2)*a(n) -5*n*(5*n-3)*a(n-1) +3*(-83*n^2+216*n-146)*a(n-2) +5*(-137*n^2+585*n-633)*a(n-3) -625*(n-3)^2*a(n-4)=0. - R. J. Mathar, Jul 27 2022

Table[Sum[Binomial[n, k]^3*2^k*3^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 30 2012 *)

(PARI) {a(n)=sum(k=0, n, binomial(n, k)^3*2^k*3^(n-k))}

for(n=0, 21, print1(a(n), ", "))

Sequence in context: A189444 A373106 A217811 * A236955 A064328 A363867

Adjacent sequences: A217277 A217278 A217279 * A217281 A217282 A217283

Paul D. Hanna, Sep 29 2012

approved