A281825 - OEIS

%I #10 Feb 21 2017 14:41:05

%S 1,1,0,-3,-2,-7,-4,-23,-6,-45,-8,655,-10,-5483,-12,929361,-14,

%T -3202321,-16,221930513,-18,-4722116559,-20,968383680659,-22,

%U -14717667114197,-24,2093660879252571,-26,-86125672563201235,-28,129848163681107301025,-30

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

%C 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 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 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 A198631 := proc(n)

%p     1/(1+exp(-x)) ;

%p     coeftayl(%,x=0,n) ;

%p     numer(%*n!) ;

%p end proc:

%p A006519 := proc(n)

%p     2^padic[ordp](n,2) ;

%p end proc:

%p L := [seq( A198631(n)/A006519(n+1),n=0..40)] ;

%p subsop(2=-1/2,L) ;

%p b := BINOMIAL(%) ;

%p for i from 1 to nops(b) do

%p     printf("%d,",numer(b[i])) ;

%p end do: # _R. J. Mathar_, Feb 21 2017

%Y Cf. A001477, A006519, A198631, A199969, A209308, A238398.

%K sign

%O 0,4

%A _Paul Curtz_, Jan 31 2017