A217282 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^k ).

1, 1, 2, 3, 5, 9, 16, 30, 57, 110, 216, 428, 857, 1730, 3516, 7191, 14785, 30544, 63370, 131976, 275811, 578219, 1215680, 2562652, 5415163, 11468455, 24338744, 51752029, 110239033, 235218046, 502674172, 1075823427, 2305661425, 4947834665, 10630848122, 22867799427

Offset: 0

Author

Paul D. Hanna, Sep 29 2012

Keywords

• nonn

Comments

Radius of convergence of g.f. A(x) is r = 0.446171506758870... where 1-r-2*r^2-2*r^3+r^4-r^5 = 0, with A(r) = (1-r^2)/(2*r^3) = 4.5087858050...

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 16*x^6 + 30*x^7 +...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2866
• Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
• Helmut Prodinger, Motzkin paths of bounded height with two forbidden contiguous subwords of length two, arXiv:2310.12497 [math.CO], 2023.

Programs

Mathematica

CoefficientList[Series[(1 - x^2 - Sqrt[(1 - x - 2*x^2 - 2*x^3 + x^4 - x^5)/(1 - x)])/(2*x^3), {x, 0, 35}], x] (* Michael De Vlieger, Oct 24 2023 *)

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)^k)+x*O(x^n))),n)}

PARI

{a(n)=polcoeff((1-x^2 - sqrt( (1-x-2*x^2-2*x^3+x^4-x^5)/(1-x +x^4*O(x^n)) ))/(2*x^3), n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: (1-x^2 - sqrt( (1-x-2*x^2-2*x^3+x^4-x^5)/(1-x) ))/(2*x^3).