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

0,3

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...

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.

Formula

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

Example

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

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 *)

Prog

(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), ", "))

Crossrefs

Sequence in context: A331966 A072176 A329700 * A047061 A136169 A047041

Adjacent sequences: A217279 A217280 A217281 * A217283 A217284 A217285

Keyword

nonn

Author

Paul D. Hanna, Sep 29 2012

Status

approved