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).

1, 2, 10, 56, 334, 2072, 13192, 85500, 561190, 3717740, 24802540, 166376256, 1120966084, 7579795628, 51408124372, 349559178116, 2382166791750, 16265392884140, 111249729804220, 762067793838960, 5227330405163524

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Robert Israel)

Formula

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

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

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

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

Mathematica

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 *)
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 *)

Prog

(Maxima)
a(n):=sum(binomial(n+(k)-1, (k))*binomial(3*n-1, n-4*k), k, 0, n);
(PARI) a(n) = sum(k=0, n, binomial(n+k-1, (k))*binomial(3*n-1, n-4*k));
vector(20, n, a(n-1)) \\ Altug Alkan, Oct 02 2015

Crossrefs

Cf. A000108, A001003, A243632.

Sequence in context: A323935 A370195 A165817 * A000172 A097971 A191277

Adjacent sequences: A243641 A243642 A243643 * A243645 A243646 A243647

Keyword

nonn

Author

Vladimir Kruchinin, Jun 08 2014

Status

approved