A217333 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k ).

1, 1, 2, 5, 12, 29, 72, 182, 466, 1207, 3158, 8334, 22158, 59299, 159614, 431838, 1173710, 3203244, 8774780, 24118522, 66497316, 183858411, 509670494, 1416231616, 3944027402, 11006186760, 30772507128, 86191006746, 241815195292, 679488418879, 1912123070998

Offset: 0

Author

Paul D. Hanna, Sep 30 2012

Keywords

• nonn

Comments

The radius of convergence of g.f. A(x) is r = 0.339332122592393190... where 1-4*r+4*r^2-4*r^3+4*r^4 = 0, with A(r) = (1-2*r)/(2*r^3) = 4.112009743749...

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 72*x^6 + 182*x^7 +...
		

Links

• Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Crossrefs

Cf. A025273, A217282.

Programs

Mathematica

(1 - 2x - Sqrt[1 - 4x + 4x^2 - 4x^3 + 4x^4])/(2x^3) + O[x]^31 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 27 2018 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k/(1-x+x*O(x^n))^k))),n)}

PARI

{a(n)=polcoeff((1-2*x - sqrt(1-4*x+4*x^2-4*x^3+4*x^4 +x^4*O(x^n)))/(2*x^3),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: (1-2*x - sqrt(1-4*x+4*x^2-4*x^3+4*x^4))/(2*x^3).

Conjecture: (n+3)*a(n) +2*(-2*n-3)*a(n-1) +4*n*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, May 17 2019

G.f. A(x) satisfies: A(x) = 1 + x * (1 + x^2*A(x)^2) / (1 - 2*x). - Ilya Gutkovskiy, Jun 30 2020