A248658 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^(2*k).

1, 1, 1, 2, 9, 28, 66, 153, 433, 1345, 3952, 10991, 30954, 90988, 271845, 804153, 2361457, 6979690, 20842285, 62493914, 187274712, 561448399, 1688263179, 5093148285, 15393417178, 46570446829, 141063389488, 427979185898, 1300470246165, 3956367018001, 12048354848013, 36728336040306

Offset: 0

Author

Paul D. Hanna, Oct 10 2014

Keywords

• nonn

Comments

Limit_{n->oo} a(n)/a(n+1) = 1 - t = t^3 = 0.3176721961... where t = ((sqrt(93)+9)/18)^(1/3) - ((sqrt(93)-9)/18)^(1/3).

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z) - (x*y*z)^3). - Ilya Gutkovskiy, Apr 23 2025

Examples

			G.f. A(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 66*x^6 + 153*x^7 +...
which equals the series:
A(x) = 1/(1-x-x^3) + 3!/1!^3*x^4/(1-x-x^3)^4 + 6!/2!^3*x^8/(1-x-x^3)^7 + 9!/3!^3*x^12/(1-x-x^3)^10 + 12!/4!^3*x^16/(1-x-x^3)^13 +...
The g.f. also equals the series:
A(x) = 1 +
x*(1 + x^2) +
x^2*(1 + 2^3*x^2 + x^4) +
x^3*(1 + 3^3*x^2 + 3^3*x^4 + x^6) +
x^4*(1 + 4^3*x^2 + 6^3*x^4 + 4^3*x^6 + x^8) +
x^5*(1 + 5^3*x^2 + 10^3*x^4 + 10^3*x^6 + 5^3*x^8 + x^10) +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A181545, A181543.

Programs

Mathematica

Table[Sum[Binomial[n-2*k,k]^3,{k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2014 *)

PARI

{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^3*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

PARI

{a(n)=polcoeff(sum(m=0, n,x^(4*m)/(1-x-x^3 +x*O(x^n))^(3*m+1)*(3*m)!/(m!)^3), n)}
for(n=0, 40, print1(a(n), ", "))

PARI

{a(n)=sum(k=0, n\3, binomial(n-2*k, k)^3)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: A(x) = Sum_{n>=0} (3*n)!/(n!)^3 * x^(4*n) / (1-x-x^3)^(3*n+1).

a(n) = Sum_{k=0..[n/3]} C(n-2*k,k)^3.

G.f.: A(x) = G( x^4/(1-x-x^3)^3 )/(1-x-x^3) where G(x) satisfies:

* G(x^3) = G( x*(1+3*x+9*x^2)/(1+6*x)^3 )/(1+6*x) and G(x) is the g.f. of A006480.