A218529 Binomial transform of -1, -1, 1, 2, -5, -16, ... (signed variant of A000111).

-1, -2, -2, 1, 4, -17, -62, 271, 1384, -7937, -50522, 353791, 2702764, -22368257, -199360982, 1903757311, 19391512144, -209865342977, -2404879675442, 29088885112831, 370371188237524, -4951498053124097, -69348874393137902, 1015423886506852351, 15514534163557086904

Offset: 0

Author

Paul Curtz, Mar 27 2013

Keywords

• sign
• less

Comments

Any distribution of signs across A000111 defines a sequence with a different binomial transform. For example, the sign pattern +--++--++--... applied to A000111 leads to A217714, and this sequence here is constructed with the sign pattern --++--++--++... .

From another point of view, we can start from this sequence, build the array of successive differences, and can read the signed variant of A000111 off the left column:

-1, -2, -2, 1, 4, -17, -62, ...

-1, 0, 3, 3, -21, -45, ...

1, 3, 0, -24, -24, ...

2, -3, -24, 0, ...

-5, -21, 24, ...

-16, 45, ...

61, ...

See the array in A163747.

The unit digits in the sequence of absolute values enter a periodic sequence 2, 1, 4, 7, 2, 1, 4, 7, ... (period 4).

Crossrefs

Cf. A000111, A163747, A217714.

Programs

Maple

seq(2^(n-1)*(euler(n,1/2)-2*euler(n,2/2)-euler(n,3/2)),n=0..24); # Peter Luschny, Feb 06 2017

Mathematica

nmax = 21; signedA111 = Table[ If[ EvenQ[ n], -EulerE[n], -(2^(n+1)*(2^(n+1) - 1)*BernoulliB[n+1])/(n+1)], {n, 0, nmax}]; Clear[t]; t[n_ , 0] := signedA111[[n+1]]; t[n_ , k_ ] := t[n, k] = t[n, k-1] + t[n+1, k-1]; a[n_] := t[0, n]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 04 2013 *)
Table[(EulerE[n] - 2^n (2 EulerE[n, 1] + EulerE[n, 3/2]))/2, {n, 0, 20}] (* Benedict W. J. Irwin, May 24 2016 *)

Formula

a(n) = A163747(n) - 1.

E.g.f.: exp(x)*(1-exp(2*x)-2*exp(x))/(1+exp(2*x)). - Philippe Deléham, Apr 01 2013

a(n) ~ n! * 2^(n+2)/Pi^(n+1) * (cos(Pi*n/2)-sin(Pi*n/2)). - Vaclav Kotesovec, Sep 24 2013

a(n) = (A122045(n) - 2^n(2*EulerE(n,1) + EulerE(n,3/2)))/2, where EulerE(n,x) is the n-th Euler polynomial. - Benedict W. J. Irwin, May 24 2016