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A281825 Numerators of the binomial transform of A198631(n)/A006519(n+1) with A198631(1) = -1 instead of 1.

%I A281825 #10 Feb 21 2017 14:41:05
%S A281825 1,1,0,-3,-2,-7,-4,-23,-6,-45,-8,655,-10,-5483,-12,929361,-14,
%T A281825 -3202321,-16,221930513,-18,-4722116559,-20,968383680659,-22,
%U A281825 -14717667114197,-24,2093660879252571,-26,-86125672563201235,-28,129848163681107301025,-30
%N A281825 Numerators of the binomial transform of A198631(n)/A006519(n+1) with A198631(1) = -1 instead of 1.
%C A281825 What is the correct name for the rational sequence c(n) = 1, -1/2, 0, -1/4, 0, 1/2, 0, -17/8, 0, 31/2, 0, ... (a variant of the second fractional Euler numbers)?
%C A281825 Its binomial transform is f(n) = 1, 1/2, 0, -3/4, -2, -7/2, -4, -23/8, -6, -45/2, -8, 655/4, -10, ... = a(n)/A006519(n+1).
%F A281825 By definition f(0) - c(0), f(1) + c(1), f(2) - c(2), f(3) + c(3), ... is an autosequence of the first kind, here 1 - 1 = 0, 1/2 - 1/2 = 0, 0 - 0 = 0, -3/4 - 1/4 = -1, -2 - 0 = -2, -7/2 + 1/2 = -3, ... i.e., t(n) = 0, 0, followed by -A001477(n), not in the OEIS, but the corresponding autosequence of the second kind is: A199969 = 0, 0, -2, -3, -4, ... Hence f(n) from c(n) and t(n).
%p A281825 A198631 := proc(n)
%p A281825     1/(1+exp(-x)) ;
%p A281825     coeftayl(%,x=0,n) ;
%p A281825     numer(%*n!) ;
%p A281825 end proc:
%p A281825 A006519 := proc(n)
%p A281825     2^padic[ordp](n,2) ;
%p A281825 end proc:
%p A281825 L := [seq( A198631(n)/A006519(n+1),n=0..40)] ;
%p A281825 subsop(2=-1/2,L) ;
%p A281825 b := BINOMIAL(%) ;
%p A281825 for i from 1 to nops(b) do
%p A281825     printf("%d,",numer(b[i])) ;
%p A281825 end do: # _R. J. Mathar_, Feb 21 2017
%Y A281825 Cf. A001477, A006519, A198631, A199969, A209308, A238398.
%K A281825 sign
%O A281825 0,4
%A A281825 _Paul Curtz_, Jan 31 2017