OFFSET
0,2
COMMENTS
Main diagonal of iterated partial sums array of central binomial coefficients (starting with the first partial sums).
FORMULA
a(n) = [x^n] 1/(sqrt(1 - 4*x)*(1 - x)^(n+1)).
a(n) = [x^n] 1/((1 - x)^(n+1)*(1 - 2*x/(1 - x/(1 - x/(1 - x/(1 - ...)))))), a continued fraction.
a(n) = 4^n*Gamma(n+1/2)*2F1(-n,n+1; 1/2-n; 1/4)/(sqrt(Pi)*Gamma(n+1)).
From Vaclav Kotesovec, Oct 16 2017: (Start)
D-finite with recurrence: 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
a(n) ~ 2^(4*n + 3/2) / (3^(n + 1/2) * sqrt(Pi*n)).
(End)
G.f.: 1/(sqrt(2*sqrt(1-4*x)-1)*sqrt(1-4*x)). - Alexander M. Haupt, Jun 24 2018
MATHEMATICA
Table[Sum[Binomial[2 k, k] Binomial[2 n - k, n], {k, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/(Sqrt[1 - 4 x] (1 - x)^(n + 1)), {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 - 2 x/(1 + ContinuedFractionK[-x, 1, {k, 1, n}])), {x, 0, n}], {n, 0, 25}]
CoefficientList[Series[1/(Sqrt[2 Sqrt[1-4 x]-1] Sqrt[1-4 x]), {x, 0, 25}], x] (* Alexander M. Haupt, Jun 24 2018 *)
PROG
(GAP) A293490 := Concatenation([1], List([1..3*10^2], n -> Sum([0..n], k -> Binomial(2*k, k)*(Binomial(2*n - k, n))))); # Muniru A Asiru, Oct 15 2017
(PARI) a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k, n)); \\ Michel Marcus, Oct 15 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 10 2017
STATUS
approved