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A129826 Transformed Bernoulli twin numbers. 9
1, -1, -2, -4, -4, 24, 120, -960, -12096, 120960, 3024000, -36288000, -1576143360, 22066007040, 1525620096000, -24409921536000, -2522591034163200, 45406638614937600, 6686974460694528000, -133739489213890560000, -27033456071346536448000, 594736033569623801856000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

We define Bernoulli twin numbers C(n) via Bernoulli numbers B(n)=A027641(n)/A027642(n) as C(0)=1, 2C(1)=-1, 3C(2)=-1, C(2n-1)= -B(2n-2) and C(2n)=B(2n), n>1. The sequence is defined as a(n)=(n+1)!*C(n).

LINKS

Table of n, a(n) for n=0..21.

FORMULA

C(n) = A051718(n)/A051717(n).

E.g.f.: Sum(n>=0)C(n) x^n/n! = 1 + x - x^2/2 + sum_{n>=1}[B(n)-B(n-1)]x^n/n! = x - x^2/2 + x/(e^x-1) - integral_{y=0..x}((y dy)/(e^y-1)).

EXAMPLE

C(0)=1; C(1)= -1/2; C(2)= -1/3; C(3)= -1/6; C(4)= -1/30; C(5)=1/30.

MATHEMATICA

c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1]=-1/2; c[2]=-1/3; a[n_] := (n+1)!*c[n]; Table[a[n], {n, 0, 21}] (* Jean-Fran├žois Alcover, Aug 08 2012 *)

CROSSREFS

Cf. A129378, A140351.

Sequence in context: A155725 A103973 A322635 * A009296 A068554 A092524

Adjacent sequences:  A129823 A129824 A129825 * A129827 A129828 A129829

KEYWORD

sign

AUTHOR

Paul Curtz, May 20 2007

EXTENSIONS

Edited and extended by R. J. Mathar, Aug 06 2008

STATUS

approved

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Last modified November 18 14:37 EST 2019. Contains 329262 sequences. (Running on oeis4.)