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A109573 E.g.f.: 2*x/(1+exp(-2*x)). 1
0, 1, 2, 0, -8, 0, 96, 0, -2176, 0, 79360, 0, -4245504, 0, 313155584, 0, -30460116992, 0, 3777576173568, 0, -581777702256640, 0, 108932957168730112, 0, -24370173276164456448, 0, 6419958484945407574016, 0, -1967044844910430876860416, 0, 693575525634287935244206080, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

(-1)^n*a(n) = 2^n*(2^n-1)*Bernoulli(n), n >= 1, are integers divisible by n. See the Koecher reference, p. 175,  Satz, with a proof, and the J. Worpitzky and L. Kronecker JraM references 94 (1883) 203-232 and 268-269, respectively. - Wolfdieter Lang, Mar 09 2017

REFERENCES

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, p. 175

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..332

FORMULA

a(0) = 0 and for n > 0 a(n) = n 2^(n-1) E_{n-1}(1) where E_{m}(x) are the Euler polynomials. - Peter Luschny, Jan 26 2009

From Sergei N. Gladkovskii, Apr 04, Nov 23 and Dec 27 2013: (Start)

Continued fractions:

E.g.f.: 2*x/(Q(0)+1) where Q(k) = 1 - 2*x/(2*k+1 - x*(2*k+1)/(x - (k+1)/Q(k+1))).

E.g.f.: 1+2*x + T(0), where T(k) = 4*k-1 - x/(1 + x/( 4*k+1 - x/(1 + x/T(k+1)))).

E.g.f.: 1 + T(0), where T(k) = 4*k-1 + x/(1 - x/( 4*k+1 + x/(1 - x/T(k+1)))).

E.g.f.: x^2*E(0)+x, where E(k) = 1 - x^2/(x^2 + (2*k+1)*(2*k+3)/E(k+1)).

(End)

a(n) = (-1)^n*2^n*(2^n-1)*Bernoulli(n), with Bernoulli(n) = A027641(n)/A027642(n). For |a(n)| see the Koecher reference, p. 175, and a comment above. - Wolfdieter Lang, Mar 09 2017

MAPLE

G:=2*x/(1+exp(-2*x)): Gser:=series(G, x=0, 35): 0, seq(n!*coeff(Gser, x^n), n=1..32); # yields the signed sequence

A109573 := n -> `if`(n = 0, 0, n*2^(n-1)*euler(n-1, 1)): # Peter Luschny, Jan 26 2009

MATHEMATICA

g[x_] = x/(-1 + Sum[(-2)^(n - 1)*x^n/n!, {n, 1, Infinity}]) h[x_, n_] = Dt[g[x], {x, n}] a[x_] = Table[h[x, n], {n, 0, 50}]; b = Abs[a[0]]

X[m_] := m Sum[(-2)^(m-1-k) k! StirlingS2[m-1, k], {k, 0, m-1}]; Table[X[i], {i, 0, 20}] (* Peter Luschny, Apr 29 2009 *)

PROG

(PARI) for(n=0, 31, print1(n*sum(k=0, n-1, (-2)^(n - 1- k) * k! * stirling(n - 1, k, 2)), ", ")) \\ Indranil Ghosh, Mar 10 2017

CROSSREFS

Cf. A027641, A027642.

Sequence in context: A211163 A239275 A186745 * A305809 A159810 A199268

Adjacent sequences:  A109570 A109571 A109572 * A109574 A109575 A109576

KEYWORD

sign,easy

AUTHOR

Roger L. Bagula, Jun 27 2005

STATUS

approved

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Last modified May 23 19:07 EDT 2019. Contains 323528 sequences. (Running on oeis4.)