cp's OEIS Frontend

A217666 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-4*x)^k.

%I A217666 #7 Dec 19 2024 17:34:12
%S A217666 1,1,2,9,43,198,903,4121,18840,86255,395397,1814662,8337729,38350063,
%T A217666 176574336,813785593,3753980313,17332179596,80089232683,370370470791,
%U A217666 1714045215632,7938075605697,36787429315319,170592514889814,791557946825363,3674974608196665
%N A217666 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-4*x)^k.
%C A217666 Radius of convergence of g.f. A(x) is |x| < 0.2116085881629750...
%C A217666 More generally, given
%C A217666 A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
%C A217666 A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).
%F A217666 G.f.: (1-4*x) / sqrt(1 - 10*x + 31*x^2 - 34*x^3 + 25*x^4).
%e A217666 G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 43*x^4 + 198*x^5 + 903*x^6 + 4121*x^7 +...
%e A217666 where the g.f. equals the series:
%e A217666 A(x) = 1 +
%e A217666 x*(1 + x/(1-4*x)) +
%e A217666 x^2*(1 + 2^2*x/(1-4*x) + x^2/(1-4*x)^2) +
%e A217666 x^3*(1 + 3^2*x/(1-4*x) + 3^2*x^2/(1-4*x)^2 + x^3/(1-4*x)^3) +
%e A217666 x^4*(1 + 4^2*x/(1-4*x) + 6^2*x^2/(1-4*x)^2 + 4^2*x^3/(1-4*x)^3 + x^4/(1-4*x)^4) +
%e A217666 x^5*(1 + 5^2*x/(1-4*x) + 10^2*x^2/(1-4*x)^2 + 10^2*x^3/(1-4*x)^3 + 5^2*x^4/(1-4*x)^4 + x^5/(1-4*x)^5) +...
%o A217666 (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-4*x +x*O(x^n))^k )), n)}
%o A217666 for(n=0,40,print1(a(n),", "))
%Y A217666 Cf. A217661, A217664, A217665.
%K A217666 nonn
%O A217666 0,3
%A A217666 _Paul D. Hanna_, Oct 10 2012