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A336729 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 + 3 * x * A(x)).

%I A336729 #31 Dec 04 2020 12:21:20
%S A336729 1,1,-2,1,10,-38,28,289,-1262,1054,11044,-51302,45604,482068,-2319176,
%T A336729 2140129,22753378,-111964106,105927508,1130780062,-5652760340,
%U A336729 5444054956,58291068808,-294808277414,287740874260,3088109246572,-15758505143192,15541351662484,167103084713608
%N A336729 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 + 3 * x * A(x)).
%H A336729 Seiichi Manyama, <a href="/A336729/b336729.txt">Table of n, a(n) for n = 0..1000</a>
%F A336729 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-3)^(n-k) * binomial(n,k) * binomial(n,k-1) for n > 0.
%F A336729 G.f.: 2/(1 - 4*x + sqrt(1 + 4*x + 16*x^2)).
%F A336729 a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(n,k) * binomial(n+k,n)/(k+1).
%F A336729 (n+1) * a(n) = -2 * (2*n-1) * a(n-1) - 16 * (n-2) * a(n-2) for n>1. - _Seiichi Manyama_, Aug 08 2020
%F A336729 a(n) ~ 2^(2*n - 1/2) * ((sqrt(3) + 1)*sin(2*Pi*n/3) + (sqrt(3) - 1)*cos(2*Pi*n/3)) / (3^(3/4) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 04 2020
%t A336729 a[0] = 1; a[n_] := Sum[(-3)^(n - k) * Binomial[n, k] * Binomial[n , k - 1], {k, 1, n}] / n; Array[a, 29, 0] (* _Amiram Eldar_, Aug 02 2020 *)
%o A336729 (PARI) {a(n) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+3*x*A)); polcoef(A, n)}
%o A336729 (PARI) {a(n) = if(n==0, 1, sum(k=1, n, (-3)^(n-k)*binomial(n, k)*binomial(n, k-1))/n)}
%o A336729 (PARI) N=40; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*x+16*x^2)))
%o A336729 (PARI) {a(n) = sum(k=0, n, (-3)^k*4^(n-k)*binomial(n, k)*binomial(n+k, n)/(k+1))}
%Y A336729 Column k=3 of A336727.
%Y A336729 Cf. A007564, A116091.
%K A336729 sign
%O A336729 0,3
%A A336729 _Seiichi Manyama_, Aug 02 2020