A317618 Expansion of e.g.f. sqrt((1 - x)/(1 - 3*x)).

1, 1, 5, 39, 417, 5685, 94365, 1847475, 41686785, 1065288105, 30411314325, 959236098975, 33129890726625, 1243507150410525, 50401090111697325, 2193907232242600875, 102075654396429338625, 5055304328553234380625, 265522264682686831945125, 14742355948224269570580375

Offset

0,3

Comments

Lah transform of A001147.

Links

Robert Israel, Table of n, a(n) for n = 0..380

N. J. A. Sloane, Transforms

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*(2*k-1)!!*n!/k!.

a(n) ~ 2 * 3^(n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 26 2019

D-finite with recurrence: (3*n^2 + 3*n)*a(n) + (-5 - 4*n)*a(n + 1) + a(n + 2)=0. - Robert Israel, Mar 26 2019

Maple

a:=series(sqrt((1 - x)/(1 - 3*x)), x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 19; CoefficientList[Series[Sqrt[(1 - x)/(1 - 3*x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] (2 k - 1)!! n!/k!, {k, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! Hypergeometric2F1[3/2, 1 - n, 2, -2], {n, 19}]]

Prog

(PARI) my(x='x + O('x^25)); Vec(serlaplace(sqrt((1 - x)/(1 - 3*x)))) \\ Michel Marcus, Mar 26 2019

Crossrefs

Cf. A000246, A001147, A002866, A052563, A084262.

Sequence in context: A070767 A124549 A308939 * A024216 A127189 A121354

Adjacent sequences: A317615 A317616 A317617 * A317619 A317620 A317621

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 01 2018

Status

approved