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A306519 Expansion of 2/(1 + 2*x + sqrt(1 - 4*x*(1 + x))). 0

%I

%S 1,0,2,4,16,56,216,848,3424,14080,58816,248832,1064064,4591744,

%T 19970432,87448832,385226240,1705979904,7590632448,33916934144,

%U 152128126976,684702330880,3091429158912,13997970530304,63550155145216,289216809762816,1319185060069376,6029646893252608

%N Expansion of 2/(1 + 2*x + sqrt(1 - 4*x*(1 + x))).

%C Inverse binomial transform of A001003.

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A001003(k).

%F a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 23 2019

%t nmax = 27; CoefficientList[Series[2/(1 + 2 x + Sqrt[1 - 4 x (1 + x)]), {x, 0, nmax}], x]

%t Table[Sum[(-1)^(n - k) Binomial[n, k] Hypergeometric2F1[1 - k, -k, 2, 2], {k, 0, n}], {n, 0, 27}]

%Y Cf. A001003, A005043, A052709, A118376, A174347.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Feb 21 2019

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Last modified June 17 15:28 EDT 2019. Contains 324194 sequences. (Running on oeis4.)