A278689 - OEIS

%I #21 Apr 03 2017 16:15:39

%S 0,1,4,25,170,1204,8736,64416,480480,3615040,27382784,208539136,

%T 1595216896,12247746560,94330470400,728474664960,5638832087040,

%U 43737154928640,339856038297600,2645063771750400,20615846154731520,160889637246074880,1257082279931412480

%N a(n) = Sum_{k=0..n} binomial(n+k,n)*binomial(2*n-3,n-k-1) for n>1, a(n) = n for n<=1.

%H Alois P. Heinz, <a href="/A278689/b278689.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (sqrt(1-8*x)*(2*x-1)+10*x+1)/(16*sqrt(1-8*x)).

%F a(n) ~ 9*8^(n-2)/sqrt(Pi*n). - _Ilya Gutkovskiy_, Nov 26 2016

%p a:= proc(n) option remember; `if`(n<3, n^2,

%p       (9*n-2)*(8*n-12)*a(n-1)/((9*n-11)*n))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 26 2016

%t CoefficientList[Series[(Sqrt[1 - 8 x] (2 x - 1) + 10 x + 1) / (16 Sqrt[1 - 8 x]), {x, 0, 30}], x] (* _Vincenzo Librandi_, Nov 26 2016 *)

%t a[n_] := Binomial[2n-3, n-1] Hypergeometric2F1[1-n, n+1, n-1, -1]; a[0]=0;

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 03 2017 *)

%o (Maxima)

%o taylor((sqrt(1-8*x)*(2*x-1)+10*x+1)/(16*sqrt(1-8*x)),x,0,10);

%o a(n):=sum(binomial(n+k,n)*binomial(2*n-3,n-k-1),k,0,n);

%o (PARI) a(n)=sum(k=0,n, binomial(n+k, n)*binomial(2*n-3, n-k-1)) \\ _Michel Marcus_, Nov 27 2016

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Nov 26 2016