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
KEYWORD
sign,easy
AUTHOR
Roger L. Bagula, Jun 27 2005
STATUS
approved