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A296771 Row sums of A050157.

%I A296771 #20 Dec 22 2017 09:23:07
%S A296771 1,3,13,58,257,1126,4882,20980,89497,379438,1600406,6720748,28117498,
%T A296771 117254268,487589572,2022568168,8371423177,34581780478,142605399982,
%U A296771 587138954428,2413944555742,9911778919348,40650232625212,166534680737368,681576405563722
%N A296771 Row sums of A050157.
%H A296771 Robert Israel, <a href="/A296771/b296771.txt">Table of n, a(n) for n = 0..1657</a>
%F A296771 a(n) = Sum_{k=0..n} (binomial(2*n, n) - binomial(2*n, n+k+1)).
%F A296771 a(n) = 2^(2*n-1)*(((n-1/2)!*(2*n+3))/(sqrt(Pi)*n!) - 1).
%F A296771 a(n) ~ 4^n*(sqrt(n/Pi) - 1/2).
%F A296771 a(n) = A037965(n+1) - A000346(n-1) for n >= 1.
%F A296771 From _Robert Israel_, Dec 21 2017: (Start)
%F A296771 a(n) = (n+3/2)*binomial(2*n,n) - 2^(2*n-1).
%F A296771 G.f.: (3/2-4*x)*(1-4*x)^(-3/2) - (1/2)*(1-4*x)^(-1).
%F A296771 64*(n+1)*(2*n+1)*a(n)-8*(2*n+3)*(5*n+4)*a(n+1)+2*(n+2)*(8*n+11)*a(n+2)-(n+3)*(n+2)*a(n+3)=0. (End)
%p A296771 A296771 := n -> add(binomial(2*n, n) - binomial(2*n, n+k+1), k=0..n):
%p A296771 seq(A296771(n), n=0..24);
%t A296771 a[n_] := 4^n ((n - 1/2)! (2 n + 3)/(2 Sqrt[Pi] n!) - 1/2);
%t A296771 Table[a[n], {n, 0, 24}]
%o A296771 (PARI) a(n) = sum(k=0, n, binomial(2*n, n) - binomial(2*n, n+k+1)) \\ _Iain Fox_, Dec 21 2017
%Y A296771 Cf. A000346, A037965, A050157, A296770.
%K A296771 nonn
%O A296771 0,2
%A A296771 _Peter Luschny_, Dec 21 2017