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 A002039 M2465 N0979 #41 Feb 16 2025 08:32:25
%S A002039 1,3,5,10,25,64,160,390,940,2270,5515,13440,32735,79610,193480,470306,
%T A002039 1143585,2781070,6762990,16445100,39987325,97232450,236432060,
%U A002039 574915770,1397981470,3399360474,8265943685,20099618590,48874630750
%N A002039 Convolution inverse of A143348.
%C A002039 Gandhi denotes f(-x) by Phi(x) and a(n) by G(n).
%D A002039 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002039 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002039 Vincenzo Librandi, <a href="/A002039/b002039.txt">Table of n, a(n) for n = 0..1000</a>
%H A002039 J. M. Gandhi, <a href="http://www.jstor.org/stable/2317132">On numbers related to partitions of a number</a>, Amer. Math. Monthly, 76 (1969), 1033-1036.
%H A002039 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Function.</a>
%F A002039 G.f.: -x / (Sum_{k>0} k * (-x)^k / (1 - (-x)^k)) = 1 / (log( f(x) )') where f(-x) = Product_{k>0} (1 - x^k) is one of Ramanujan's theta functions. - _Michael Somos_, Apr 08 2003
%F A002039 a(n) ~ c * d^n, where d = -1/A143441 = 2.43161993449532399475429572773256778... and c = 0.765603960074106532799232452562411022387973764575133091283490410339311... - _Vaclav Kotesovec_, Jun 02 2018
%F A002039 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma(k+1) * a(n-k). - _Ilya Gutkovskiy_, May 27 2020
%e A002039 1 + 3*x + 5*x^2 + 10*x^3 + 25*x^4 + 64*x^5 + 160*x^6 + 390*x^7 + 940*x^8 + ...
%t A002039 max = 28; f[x_] := -x / Sum[ k*(-x)^k/(1-(-x)^k), {k, 1, max+1}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* _Jean-François Alcover_, Nov 07 2011, after _Michael Somos_ *)
%o A002039 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / log( eta( -x + x^2 * O(x^n)))', n))} /* _Michael Somos_, Apr 05 2003 */
%Y A002039 Cf. A002040, A143348.
%K A002039 nonn,nice,easy
%O A002039 0,2
%A A002039 _N. J. A. Sloane_, _Simon Plouffe_