A318223 Expansion of e.g.f. exp(x/(1 + 2*x)).

1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151, 236126701, -6169406979, 161388751657, -4327824442967, 120012465557349, -3450029411174219, 102741264191105761, -3160671409312412703, 99982488984008583133, -3230094912866216253971, 105481073534842477321881, -3423260541695907002392679

Offset: 0

Author

Ilya Gutkovskiy, Aug 21 2018

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A002866, A025168, A111884, A122803.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x/(1+2*x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 07 2019

Maple

seq(n!*coeff(series(exp(x/(1+2*x)),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 21; CoefficientList[Series[Exp[x/(1 + 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-2)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-2)^(k - 1) k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Join[{1}, Table[(-2)^(n - 1) n! Hypergeometric1F1[1 - n, 2, 1/2], {n, 21}]]

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1+2*x)))) \\ G. C. Greubel, Feb 07 2019

Sage

m = 30; T = taylor(exp(x/(1+2*x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 07 2019

Formula

E.g.f.: Product_{k>=1} exp((-2)^(k-1)*x^k).

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

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