cp's OEIS Frontend

A073592 Euler transform of negative integers.

%I A073592 #39 Nov 11 2020 08:28:16
%S A073592 1,-1,-2,-1,0,4,4,7,3,-2,-9,-17,-25,-24,-13,-1,32,61,97,111,112,74,8,
%T A073592 -108,-243,-392,-512,-569,-542,-358,-33,473,1078,1788,2395,2865,2955,
%U A073592 2569,1496,-245,-2751,-5783,-9121,-12299,-14739,-15806,-14719,-10930,-3813,6593,20284,36139,53081,68620,80539
%N A073592 Euler transform of negative integers.
%C A073592 1/A(x) is g.f. for A000219.
%H A073592 Seiichi Manyama, <a href="/A073592/b073592.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vaclav Kotesovec)
%H A073592 E. M. Wright, <a href="http://dx.doi.org/10.1093/qmath/17.1.39">Coefficients of a reciprocal generating function</a>, Quart. J. Math. 17 (1) (1966) 39-43, <a href="http://adsabs.harvard.edu/abs/1966QJMat..17...39W">ADS Abstracts</a>.
%H A073592 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A073592 G.f.: Product_{k>0} (1-x^k)^k.
%F A073592 a(n) = -1/n*Sum_{k=1..n} sigma[2](k)*a(n-k).
%F A073592 G.f.: exp( Sum_{n>=1} -sigma_2(n)*x^n/n ). - _Seiichi Manyama_, Mar 04 2017
%p A073592 a:= proc(n) option remember; `if`(n=0, 1, -add(
%p A073592       numtheory[sigma][2](j)*a(n-j), j=1..n)/n)
%p A073592     end:
%p A073592 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 12 2015
%t A073592 nmax=50; CoefficientList[Series[Exp[Sum[-x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 02 2015 *)
%t A073592 a[n_]:= a[n] = -1/n*Sum[DivisorSigma[2,k]*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* _Vaclav Kotesovec_, Mar 02 2015 *)
%o A073592 (SageMath) # uses[EulerTransform from A166861]
%o A073592 b = EulerTransform(lambda n: -n)
%o A073592 print([b(n) for n in range(55)]) # _Peter Luschny_, Nov 11 2020
%Y A073592 Column k=1 of A283272.
%Y A073592 Cf. A000219, A001157, A001478, A026007, A156616, A255528.
%K A073592 sign
%O A073592 0,3
%A A073592 _Vladeta Jovovic_, Aug 28 2002