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a(n) = Sum_{k=0..n} binomial(2*k, k) * binomial(2*k+1,k).
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%I #18 Apr 18 2023 15:59:13

%S 1,7,67,767,9587,126011,1711595,23796515,336666215,4828084575,

%T 69994481871,1023793569567,15086216016367,223704570996367,

%U 3335098322412367,49954148031128767,751296616443141667

%N a(n) = Sum_{k=0..n} binomial(2*k, k) * binomial(2*k+1,k).

%C Old name was "A binomial sum".

%H Vincenzo Librandi, <a href="/A082578/b082578.txt">Table of n, a(n) for n = 0..200</a>

%F Recurrence: (n+3)*(n+2)*a(n+2) - (17*n^2+69*n+66)*a(n+1) + (16*n^2+64*n+60)*a(n) = 0.

%F a(n) ~ 2^(4*n+5)/(15*Pi*n). - _Vaclav Kotesovec_, Oct 14 2012

%t Table[Sum[Binomial[2k,k]*Binomial[2k+1,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)

%o (Maxima) makelist(sum(binomial(2*k,k)*binomial(2*k+1,k),k,0,n),n,0,12);

%Y Partial sums of A000894.

%K easy,nonn

%O 0,2

%A _Emanuele Munarini_, May 07 2003

%E Name changed by _Wesley Ivan Hurt_, Apr 18 2023