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

1, 1, 0, -3, -2, -7, -4, -23, -6, -45, -8, 655, -10, -5483, -12, 929361, -14, -3202321, -16, 221930513, -18, -4722116559, -20, 968383680659, -22, -14717667114197, -24, 2093660879252571, -26, -86125672563201235, -28, 129848163681107301025, -30

Offset

0,4

Comments

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)?

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).

Links

Table of n, a(n) for n=0..32.

Formula

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).

Maple

A198631 := proc(n)
    1/(1+exp(-x)) ;
    coeftayl(%, x=0, n) ;
    numer(%*n!) ;
end proc:
A006519 := proc(n)
    2^padic[ordp](n, 2) ;
end proc:
L := [seq( A198631(n)/A006519(n+1), n=0..40)] ;
subsop(2=-1/2, L) ;
b := BINOMIAL(%) ;
for i from 1 to nops(b) do
    printf("%d, ", numer(b[i])) ;
end do: # R. J. Mathar, Feb 21 2017

Crossrefs

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

Sequence in context: A324867 A265386 A075627 * A011384 A128140 A213579

Adjacent sequences: A281822 A281823 A281824 * A281826 A281827 A281828

Keyword

sign

Author

Paul Curtz, Jan 31 2017

Status

approved