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URL: https://oeis.org/A217282

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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 ).
5
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
Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
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 * A390425 A047061 A136169
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 29 2012
STATUS
approved