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 A239275 a(n) = numerator(2^n * Bernoulli(n, 1)). 3
 1, 1, 2, 0, -8, 0, 32, 0, -128, 0, 2560, 0, -1415168, 0, 57344, 0, -118521856, 0, 5749735424, 0, -91546451968, 0, 1792043646976, 0, -1982765704675328, 0, 286994513002496, 0, -3187598700536922112, 0, 4625594563496048066560, 0, -16555640873195841519616, 0, 22142170101965089931264, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Difference table of f(n) = 2^n *A164555(n)/A027642(n) = a(n)/A141459(n): 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. Main diagonal: A212196(n)/A181131(n). See A190339(n). First upper diagonal: A229023(n)/A181131(n). 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. Sum of the antidiagonals: 1,1,0,-1,0,3,0,-17,... = (-1)^n*A036968(n) = -A226158(n+1). 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 a(n) = numerators of 2^n *A164555(n)/A027642(n). Numerators of the binomial transform of A157779(n)/(interleave A001897(n), 1)(conjectured). 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 Cf. A141459 (denominators), A001896/A001897, A027641/A027642. Sequence in context: A199573 A103424 A211163 * A186745 A109573 A305809 Adjacent sequences:  A239272 A239273 A239274 * A239276 A239277 A239278 KEYWORD sign,frac,easy AUTHOR Paul Curtz, Mar 13 2014 STATUS approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)