This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307067 #12 Jan 25 2024 04:56:23 %S A307067 1,0,1,1,2,4,6,12,19,36,60,108,187,328,576,1005,1765,3084,5408,9461, %T A307067 16575,29017,50812,88977,155792,272813,477684,836466,1464654,2564685, %U A307067 4490833,7863610,13769463,24110774,42218847,73926591,129448088,226667986,396903536,694991728 %N A307067 Expansion of 1/(2 - Product_{k>=2} (1 + x^k)). %C A307067 Invert transform of A025147. %H A307067 G. C. Greubel, <a href="/A307067/b307067.txt">Table of n, a(n) for n = 0..1000</a> %F A307067 a(0) = 1; a(n) = Sum_{k=1..n} A025147(k)*a(n-k). %F A307067 From _G. C. Greubel_, Jan 24 2024: (Start) %F A307067 G.f.: (1+x)/(2*(1+x) - QP(x^2)/QP(x)), where QP(x) = QPochhammer(x). %F A307067 G.f.: (1+x)/(2*(1+x) - x^(1/24)*eta(x^2)/eta(x)), where eta(x) is the Dedekind eta function. (End) %p A307067 a:=series(1/(2-mul((1+x^k),k=2..100)),x=0,40): seq(coeff(a,x,n),n=0..39); # _Paolo P. Lava_, Apr 03 2019 %t A307067 nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x] %o A307067 (Magma) %o A307067 m:=80; %o A307067 R<x>:=PowerSeriesRing(Integers(), m); %o A307067 Coefficients(R!( 1/(2 - (&*[1+x^j: j in [2..m+2]])) )); // _G. C. Greubel_, Jan 24 2024 %o A307067 (SageMath) %o A307067 m=80; %o A307067 def f(x): return 1/( 2 - product(1+x^j for j in range(2, m+3)) ) %o A307067 def A307067_list(prec): %o A307067 P.<x> = PowerSeriesRing(QQ, prec) %o A307067 return P( f(x) ).list() %o A307067 A307067_list(m) # _G. C. Greubel_, Jan 24 2024 %Y A307067 Cf. A025147, A055887, A299106, A304969, A307057, A307060, A317536. %K A307067 nonn %O A307067 0,5 %A A307067 _Ilya Gutkovskiy_, Mar 22 2019