A320767 Inverse Euler transform applied once to {1,-1,0,0,0,...}, twice to {1,0,0,0,0,...}, or three times to {1,1,1,1,1,...}.

1, 1, -2, 1, -1, 2, -3, 4, -5, 8, -13, 18, -25, 40, -62, 90, -135, 210, -324, 492, -750, 1164, -1809, 2786, -4305, 6710, -10460, 16264, -25350, 39650, -62057, 97108, -152145, 238818, -375165, 589520, -927200, 1459960, -2300346, 3626200, -5720274, 9030450

Offset: 0

Author

Gus Wiseman, Oct 20 2018

Keywords

• sign

Comments

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Crossrefs

Cf. A000081, A001970, A007562, A007294, A034691, A059966, A061255, A061256, A061257, A065490, A073576, A117209.

Programs

Mathematica

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
Nest[EulerInvTransform,Array[DiscreteDelta,50,0],2]