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

%I A248658 #19 Apr 23 2025 16:19:44
%S A248658 1,1,1,2,9,28,66,153,433,1345,3952,10991,30954,90988,271845,804153,
%T A248658 2361457,6979690,20842285,62493914,187274712,561448399,1688263179,
%U A248658 5093148285,15393417178,46570446829,141063389488,427979185898,1300470246165,3956367018001,12048354848013,36728336040306
%N A248658 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^(2*k).
%C A248658 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).
%C A248658 Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z) - (x*y*z)^3). - _Ilya Gutkovskiy_, Apr 23 2025
%H A248658 Paul D. Hanna, <a href="/A248658/b248658.txt">Table of n, a(n) for n = 0..500</a>
%F A248658 G.f.: A(x) = Sum_{n>=0} (3*n)!/(n!)^3 * x^(4*n) / (1-x-x^3)^(3*n+1).
%F A248658 a(n) = Sum_{k=0..[n/3]} C(n-2*k,k)^3.
%F A248658 G.f.: A(x) = G( x^4/(1-x-x^3)^3 )/(1-x-x^3) where G(x) satisfies:
%F A248658 * 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.
%e A248658 G.f. A(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 66*x^6 + 153*x^7 +...
%e A248658 which equals the series:
%e A248658 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 +...
%e A248658 The g.f. also equals the series:
%e A248658 A(x) = 1 +
%e A248658 x*(1 + x^2) +
%e A248658 x^2*(1 + 2^3*x^2 + x^4) +
%e A248658 x^3*(1 + 3^3*x^2 + 3^3*x^4 + x^6) +
%e A248658 x^4*(1 + 4^3*x^2 + 6^3*x^4 + 4^3*x^6 + x^8) +
%e A248658 x^5*(1 + 5^3*x^2 + 10^3*x^4 + 10^3*x^6 + 5^3*x^8 + x^10) +...
%t A248658 Table[Sum[Binomial[n-2*k,k]^3,{k,0,Floor[n/3]}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2014 *)
%o A248658 (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)}
%o A248658 for(n=0, 40, print1(a(n), ", "))
%o A248658 (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)}
%o A248658 for(n=0, 40, print1(a(n), ", "))
%o A248658 (PARI) {a(n)=sum(k=0, n\3, binomial(n-2*k, k)^3)}
%o A248658 for(n=0, 40, print1(a(n), ", "))
%Y A248658 Cf. A181545, A181543.
%K A248658 nonn
%O A248658 0,4
%A A248658 _Paul D. Hanna_, Oct 10 2014