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Numerator of the n-th term of the inverse Binomial Transform of the Bernoulli sequence prefixed with 0.
0

%I #8 Jan 12 2019 20:02:51

%S 0,1,-5,14,-23,349,-499,793,-1038,7901,-9791,65488,-78193,795259,

%T -925389,1615811,-1841036,67142767,-75821437,358067518,-388783203,

%U -521129621,480390923,133108162049

%N Numerator of the n-th term of the inverse Binomial Transform of the Bernoulli sequence prefixed with 0.

%C The inverse binomial transform of 0, 1, -1/2, 1/6, 0, ... is A(n) = 0, 1, -5/2, 14/3, -23/3, ... The current sequence is defined by the numerators; the denominators are A100650(n).

%C There is a connection to the sequence b(n) = 0, 1, 1/2, 1/6, 0, -1/30, ... of modified Bernoulli numbers [b(0)=0, b(2) = -Bernoulli(1), b(n) = Bernoulli(n-1) if n <> 2] discussed in A165142: The inverse binomial transform of b(n) is c(n) = 0, 1, -3/2, 5/3, -5/3, 49/30, -49/30, ..., and c(n) - A(n) = (-1)^n*A000217(n-1).

%p read("transforms") ;

%p A174264 := proc(n) local b; b := [0,seq(bernoulli(i),i=0..n+1)] ; BINOMIALi(b) ; numer(op(n+1,%)) ; end proc:

%p seq(A174264(n),n=0..30) ; # _R. J. Mathar_, Jan 21 2011

%Y Cf. A164558.

%K sign,frac

%O 0,3

%A _Paul Curtz_, Mar 14 2010