A052563 -id:A052563 - cp's OEIS frontend

A052585 E.g.f. 1/(1-x-2*x^2).

1, 1, 6, 30, 264, 2520, 30960, 428400, 6894720, 123742080, 2478470400, 54486432000, 1308153369600, 34005760588800, 952248474777600, 28566146568960000, 914137612996608000, 31080323154456576000, 1118898035934142464000, 42518003720397004800000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Laguerre transform is A052563. - Paul Barry, Aug 08 2008

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 530

Cf. A080599, A005442.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1 -x -2*x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

spec := [S,{S=Sequence(Union(Z,Prod(Z,Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{m = 50}, CoefficientList[Series[-1/(-1 + x + 2*x^2), {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 17 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1 -x -2*x^2))) \\ G. C. Greubel, May 17 2018

E.g.f.: 1/(1 -x -2*x^2).
Recurrence: a(1)=1, a(0)=1, (-2*n^2-6*n-4)*a(n)+(-2-n)*a(n+1)+a(n+2)=0.
a(n) = Sum(1/9*(1+4*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+2*_Z^2))*n!.
a(n) = n!*A001045(n+1). - Paul Barry, Aug 08 2008
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A080599 and A005442. - Peter Bala, Dec 07 2011

a(18)-a(19) added by G. C. Greubel, May 17 2018

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 01 2018

nonn

Lah transform of A001147.

Robert Israel, Table of n, a(n) for n = 0..380
N. J. A. Sloane, Transforms

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

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

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}]]

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

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

cp's OEIS Frontend

A052585 E.g.f. 1/(1-x-2*x^2).

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