A188312 Expansion of (1/(1-x^2))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.
1, 1, 4, 12, 45, 174, 709, 2978, 12825, 56303, 251060, 1133943, 5176926, 23851690, 110759081, 517853840, 2435786531, 11517940357, 54722081630, 261089977806, 1250479470053, 6009884614944, 28975052979797, 140098515402139, 679189779433800, 3300702453217325, 16076773046682690
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Crossrefs
Cf. A188314.
Programs
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Magma
m:=25; R
:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x^2 -Sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)))); // G. C. Greubel, Aug 14 2018 -
Maple
a := n -> add((-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4), i=0..n); seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018
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Mathematica
CoefficientList[Series[(1-x^2 -Sqrt[1-4*x-6*x^2+x^4])/(2*x*(1+x)), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *) -
Maxima
a(n):=sum(sum((-1)^(n-k-i)*binomial(k+i-1, k-1)*binomial(2*k+i-2, k+i-1)* binomial(n-i-1, n-k-i)/k,k,1,n-i),i,0,n); /* Vladimir Kruchinin, May 03 2018 */
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PARI
x='x+O('x^50); Vec((1-x^2 -sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x))) \\ G. C. Greubel, Aug 14 2018
Formula
G.f.: (1-x^2 - sqrt(1-4*x-6*x^2+x^4))/(2*x*(1+x)).
G.f.: u(x)=1/(1-x^2-x/(1-x-x*u(x))).
G.f.: 1/(1-x^2-x/(1-x-x/(1-x^2-x/(1-x-x/(1-...))))) (continued fraction).
Conjecture: (n+1)*a(n) +(3-4*n)*a(n-1) + (7-6*n)*a(n-2) -a(n-3) +(n-4)*a(n-4)=0. - R. J. Mathar, Nov 15 2011
a(n) = a(n-1) + (-1)^n + Sum_{i=0..n-1} a(i)*a(n-1-i). - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n} Sum_{k=1..n-i} (-1)^(n-k-i)*C(k+i-1,k-1)*C(2*k+i-2,k+i-1)*C(n-i-1,n-k-i)/k. - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n} (-1)^(n-i)*hypergeom([(i+1)/2, i/2+1, i-n], [1, 2], 4). - Peter Luschny, May 03 2018
Comments