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A318223 Expansion of e.g.f. exp(x/(1 + 2*x)). 2
1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151, 236126701, -6169406979, 161388751657, -4327824442967, 120012465557349, -3450029411174219, 102741264191105761, -3160671409312412703, 99982488984008583133, -3230094912866216253971, 105481073534842477321881, -3423260541695907002392679 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

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).

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}]]

PROG

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

(MAGMA) m:=30; R<x>:=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

(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

CROSSREFS

Cf. A002866, A025168, A111884, A122803.

Sequence in context: A277922 A182189 A198447 * A162326 A122455 A126390

Adjacent sequences:  A318220 A318221 A318222 * A318224 A318225 A318226

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Aug 21 2018

STATUS

approved

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Last modified September 25 15:27 EDT 2021. Contains 347658 sequences. (Running on oeis4.)