cp's OEIS Frontend

A022661 Expansion of Product_{m>=1} (1-m*q^m).

%I A022661 #29 Sep 08 2022 08:44:46
%S A022661 1,-1,-2,-1,-1,5,1,13,4,0,2,-8,-61,-31,13,-156,21,11,223,92,91,426,
%T A022661 972,165,141,-1126,440,1294,-4684,-2755,-5748,-2414,-6679,10511,
%U A022661 -10048,-19369,19635,22629,14027,76969,-1990,40193,-10678,75795,215767,-54322,-40882
%N A022661 Expansion of Product_{m>=1} (1-m*q^m).
%C A022661 Is a(9) the only occurrence of 0 in this sequence? - _Robert Israel_, Jun 02 2015
%H A022661 Robert Israel, <a href="/A022661/b022661.txt">Table of n, a(n) for n = 0..10000</a>
%p A022661 P:= mul(1-m*q^m,m=1..100):
%p A022661 S:= series(P,q,101):
%p A022661 seq(coeff(S,q,j),j=0..100); # _Robert Israel_, Jun 02 2015
%p A022661 # second Maple program:
%p A022661 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A022661       `if`(n=0, 1, b(n, i-1)-`if`(i>n, 0, i*b(n-i, i-1))))
%p A022661     end:
%p A022661 a:= n-> b(n$2):
%p A022661 seq(a(n), n=0..60);  # _Sean A. Irvine_ (after _Alois P. Heinz_), May 19 2019
%t A022661 nmax = 40; CoefficientList[Series[Product[1 - k*x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 15 2015 *)
%t A022661 nmax = 40; CoefficientList[Series[Exp[-Sum[PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 15 2015 *)
%t A022661 (* More efficient program: *) nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = -1; Do[Do[poly[[j+1]] -= k*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 07 2016 *)
%o A022661 (PARI) m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n))) \\ _G. C. Greubel_, Feb 18 2018
%o A022661 (Magma) Coefficients(&*[(1-m*x^m):m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 18 2018
%Y A022661 Cf. A006906, A022629, A022693, A266964.
%K A022661 sign
%O A022661 0,3
%A A022661 _N. J. A. Sloane_