A293490 a(n) = Sum_{k=0..n} binomial(2*k, k)*binomial(2*n-k, n).

1, 4, 18, 84, 400, 1932, 9436, 46512, 231066, 1155660, 5813808, 29396952, 149305884, 761282032, 3894953640, 19987999696, 102847396416, 530446714812, 2741576339716, 14196136939600, 73631851898220, 382483602131400, 1989514312826400, 10361255764532400, 54020655931542300, 281933439875693424

Offset: 0

Author

Ilya Gutkovskiy, Oct 10 2017

Keywords

• nonn

Comments

Main diagonal of iterated partial sums array of central binomial coefficients (starting with the first partial sums).

Crossrefs

Cf. A000984, A002426, A006134, A026375, A270447.

Programs

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

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 *)

PARI

a(n) = sum(k=0, n, binomial(2*k, k)*binomial(2*n-k, n)); \\ Michel Marcus, Oct 15 2017

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