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A243644 Expansion of x*log'(((1-sqrt(1-4*x))/2-sqrt(((-sqrt(1-4*x)-11)*(1-sqrt(1-4*x)))/4+1)+1)/4). 2

%I #23 Feb 15 2017 03:04:27

%S 1,2,10,56,334,2072,13192,85500,561190,3717740,24802540,166376256,

%T 1120966084,7579795628,51408124372,349559178116,2382166791750,

%U 16265392884140,111249729804220,762067793838960,5227330405163524

%N Expansion of x*log'(((1-sqrt(1-4*x))/2-sqrt(((-sqrt(1-4*x)-11)*(1-sqrt(1-4*x)))/4+1)+1)/4).

%H G. C. Greubel, <a href="/A243644/b243644.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..120 from Robert Israel)

%F a(n) = Sum_{k=0..n} binomial(n+(k)-1,(k))*binomial(3*n-1,n-4*k).

%F A(x) = x*log'(x*C(x)*S(x*C(x))), where C(x) is g.f. A000108, S(x) is g.f. A001003.

%F a(n) ~ sqrt((6+2*sqrt(2))/7) * (7+5*sqrt(2))^n / (sqrt(Pi*n) * 2^(n+5/4)). - _Vaclav Kotesovec_, Jun 15 2014

%F a(n) = [x^n] ( (1 - x)^2/((1 - x)^4 - x^4) )^n. O.g.f. A(x) is essentially obtained by logarithmically differentiating the o.g.f of A243632. - _Peter Bala_, Oct 02 2015

%t CoefficientList[Series[-((2*(5 + Sqrt[1-4*x] + Sqrt[-6+10*Sqrt[1-4*x] - 4*x]) * x)/((-3 + Sqrt[1-4*x] + Sqrt[-6 + 10*Sqrt[1-4*x] - 4*x]) * Sqrt[1-4*x]*Sqrt[-6 + 10*Sqrt[1-4*x] - 4*x])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)

%t Table[Sum[Binomial[n + k - 1, k]*Binomial[3*n - 1, n - 4*k], {k, 0, n}], {n,0,50}] (* _G. C. Greubel_, Feb 14 2017 *)

%o (Maxima)

%o a(n):=sum(binomial(n+(k)-1,(k))*binomial(3*n-1,n-4*k),k,0,n);

%o (PARI) a(n) = sum(k=0, n, binomial(n+k-1,(k))*binomial(3*n-1,n-4*k));

%o vector(20, n, a(n-1)) \\ _Altug Alkan_, Oct 02 2015

%Y Cf. A000108, A001003, A243632.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Jun 08 2014

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Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)