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

0,3

Links

Table of n, a(n) for n=0..20.

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

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}]

Prog

(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

Crossrefs

Cf. A001147, A005359, A034430, A052585, A317618.

Sequence in context: A364798 A349875 A376934 * A357245 A157197 A363410

Adjacent sequences: A331400 A331401 A331402 * A331404 A331405 A331406

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 16 2020

Status

approved