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A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

%I A195240 #20 Apr 17 2019 22:47:10
%S A195240 0,1,1,7,8,11,10,7,8,19,14,337,1028,5,-2,-1681,1936,22133,-21734,
%T A195240 -87223,87388,427291,-427222,-118181363,118182728,4276553,-4276550,
%U A195240 -11874730297,11874730732,4307920641583
%N A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).
%C A195240 The array of (-1)^n*A176328(n)/A176591(n) and its first, second, etc. differences in subsequence rows starts as follows:
%C A195240 0,     1,     2,   19/6,   14/3, 199/30,    137/15, ... (-1)^n * A176328(n)/A176591(n),
%C A195240 1,     1,   7/6,    3/2,  59/30,    5/2,    127/42, ... see A176328,
%C A195240 0,   1/6,   1/3,   7/15,   8/15,  11/21,     10/21, ...
%C A195240 1/6, 1/6,  2/15,   1/15, -1/105,  -1/21,    -1/105, ... see A190339
%C A195240 0, -1/30, -1/15, -8/105, -4/105,  4/105, -116/1155, ...
%C A195240 The numerators in the 3rd row, 0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, 7/15, 8/15, 19/33, 14/33, 337/1365, 1028/1365, 5/3, -2/3, -1681/255, 1936/255, ... define the current sequence.
%C A195240 The associated denominators are 1, 6 and followed by 3, 15, 15 etc as provided in A172087.
%C A195240 The second column of the array, 1, 1, 1/6, 1/6, -1/30, -1/30, ... contains doubled A000367(n)/A002445(n). These are related to A176150, A176144, and A176184.
%C A195240 In the first subdiagonal of the array we see 1, 1/6, 2/15, -8/150, 8/105, -32/321, 6112/15015, -3712/2145 , ... continued as given by A181130 and A181131.
%H A195240 Vincenzo Librandi, <a href="/A195240/b195240.txt">Table of n, a(n) for n = 0..300</a>
%F A195240 a(2*n+1) + a(2*n+2) = A172087(2*n+2) = A172087(2*n+3), n >= 1.
%p A195240 read("transforms") ;
%p A195240 evb := [0, 1, 0, seq(bernoulli(n), n=2..30)] ;
%p A195240 ievb := BINOMIALi(evb) ;
%p A195240 [seq((-1)^n*op(n,ievb),n=1..nops(ievb))] ;
%p A195240 DIFF(%) ;
%p A195240 DIFF(%) ;
%p A195240 apply(numer,%) ; # _R. J. Mathar_, Sep 20 2011
%t A195240 evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 32}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k+1]], {k, 0, n}], {n, 0, Length[evb]-3}]; Differences[ievb, 2] // Numerator (* _Jean-François Alcover_, Sep 09 2013, after _R. J. Mathar_ *)
%K A195240 sign,frac
%O A195240 0,4
%A A195240 _Paul Curtz_, Sep 13 2011