A090336 Odd-indexed terms of the first binomial transform equals 1 and the even-indexed terms of the third binomial transform equals 1, with a(0)=1.

1, 0, -8, 24, 64, -480, -3968, 34944, 354304, -4062720, -51734528, 724568064, 11070521344, -183240744960, -3266330329088, 62382319632384, 1270842139869184, -27507470234419200, -630424777639067648, 15250953398036987904, 388362339077349965824

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: 2*(1+sinh(2*x))/(1+exp(4*x)).

a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) / (Pi/4)^(n+1). - Vaclav Kotesovec, Mar 06 2014

a(n) = 2^(n-1)*(EulerE(n) - 2^n*(EulerE(n,-1/2) - 2*EulerE(n,0))). - Benedict W. J. Irwin, May 26 2016

Example

Successive binomial transforms are:
0th: {1,0,-8,24,64,-480,-3968,34944,354304,-4062720,...}
1st: {1,1,-7,1,113,1,-5527,1,501473,1,-73163047,1,...}
2nd: {1,2,-4,-16,80,512,-3904,-34816,354560,4063232,...}
3rd: {1,3,1,-21,1,723,1,-49221,1,5746083, 1,...} and
4th: {1,4,8,-8,-64,544,3968, -34688,-354304,4063744,...}
The sum of this sequence with its 4th binomial transform equals {2,4,0,16,0,64,0,64,0,256,0,1024,...}, which has e.g.f.: 2+2sinh(2x).
This describes the e.g.f.: A+exp(4x)*A=2+2sinh(2x).

Mathematica

CoefficientList[Series[2*(1+Sinh[2*x])/(1+E^(4*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 06 2014 *)
Table[2^(n - 1)*(EulerE[n]-2^n (EulerE[n, -1/2] - 2 EulerE[n, 0])), {n, 0, 20}] (* Benedict W. J. Irwin, May 26 2016 *)

Crossrefs

Cf. A090145, A090158.

Sequence in context: A205963 A208895 A111071 * A364245 A200253 A343546

Adjacent sequences: A090333 A090334 A090335 * A090337 A090338 A090339

Keyword

sign

Author

Paul D. Hanna, Nov 25 2003

Status

approved