OFFSET
0,3
COMMENTS
1, 1, 2/3, 0, -8/15, 0, 32/21, 0,...
0, -1/3, -2/3, -8/15, 8/15, 32/21, -32/21,...
-1/3, -1/3, 2/15, 16/15, 104/105, -64/21,...
0, 7/15, 14/15, -8/105, -424/105,...
7/15, 7/15, -106/105, -416/105,...
0, -31/21, -62/31,
-31/21, -31/21,...
0,... etc.
The inverse binomial transform of f(n) is g(n). Reciprocally, the inverse binomial transform of g(n) is f(n) with -1 instead of f(1)=1, i.e., f(n) signed.
Following A211163(n+2), f(n) is the coefficients of a polynomial in Pi^n.
Bernoulli numbers, twice, and Genocchi numbers, twice, are linked to Pi.
f(n) - g(n) = -A226158(n).
Also the numerators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The denominators are A141459. - Peter Luschny, Nov 22 2015
(-1)^n*a(n) = 2^n*numerator(A027641(n)/A027642(n)) (that is the present sequence with a(1) = -1 instead of +1). - Wolfdieter Lang, Jul 05 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017. See B(2;n), eq. (53).
FORMULA
MAPLE
seq(numer(2^n*bernoulli(n, 1)), n=0..35); # Peter Luschny, Jul 17 2017
MATHEMATICA
Table[Numerator[2^n*BernoulliB[n, 1]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)
PROG
(Python)
from sympy import bernoulli
def a(n): return (2**n * bernoulli(n, 1)).numerator()
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017
CROSSREFS
KEYWORD
sign,frac,easy
AUTHOR
Paul Curtz, Mar 13 2014
STATUS
approved