A331403 E.g.f.: 1 / ((1 + x) * sqrt(1 - 2*x)).
1, 0, 3, 6, 81, 540, 7155, 85050, 1346625, 22339800, 431331075, 9004668750, 208178118225, 5199538043700, 140664514065075, 4080315642653250, 126613733680058625, 4180226398201854000, 146399020309066399875, 5419213146765629961750, 211446723837565171580625
Offset: 0
Keywords
Programs
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Mathematica
nmax = 20; CoefficientList[Series[1/((1 + x) Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]! Table[n! Sum[(-1)^(n - k) (2 k - 1)!!/k!, {k, 0, n}], {n, 0, 20}] -
PARI
a(n) = {n! * sum(k=0, n, (-1)^(n - k) * (2*k)! / (2^k*k!^2))} \\ Andrew Howroyd, Jan 16 2020 -
PARI
seq(n) = {Vec(serlaplace(1 / ((1 + x) * sqrt(1 - 2*x + O(x*x^n)))))} \\ Andrew Howroyd, Jan 16 2020
Formula
a(n) = n! * Sum_{k=0..n} (-1)^(n - k) * (2*k - 1)!! / k!.
D-finite with recurrence: a(n) +(-n+1)*a(n-1) -(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 25 2020
a(n) ~ 2^(n + 3/2) * n^n / (3*exp(n)). - Vaclav Kotesovec, Jan 26 2020