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

1, 0, 2, 4, 16, 56, 216, 848, 3424, 14080, 58816, 248832, 1064064, 4591744, 19970432, 87448832, 385226240, 1705979904, 7590632448, 33916934144, 152128126976, 684702330880, 3091429158912, 13997970530304, 63550155145216, 289216809762816, 1319185060069376, 6029646893252608

Offset: 0

Author

Ilya Gutkovskiy, Feb 21 2019

Keywords

• nonn

Comments

Inverse binomial transform of A001003.

Crossrefs

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

Programs

Mathematica

nmax = 27; CoefficientList[Series[2/(1 + 2 x + Sqrt[1 - 4 x (1 + x)]), {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Hypergeometric2F1[1 - k, -k, 2, 2], {k, 0, n}], {n, 0, 27}]

Formula

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

a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 23 2019

D-finite with recurrence: (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-4*n+5)*a(n-2) +4*(-n+2)*a(n-3)=0. - R. J. Mathar, Jan 25 2020