A002039 - OEIS

%I M2465 N0979 #40 May 27 2020 20:16:24

%S 1,3,5,10,25,64,160,390,940,2270,5515,13440,32735,79610,193480,470306,

%T 1143585,2781070,6762990,16445100,39987325,97232450,236432060,

%U 574915770,1397981470,3399360474,8265943685,20099618590,48874630750

%N Convolution inverse of A143348.

%C Gandhi denotes f(-x) by Phi(x) and a(n) by G(n).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002039/b002039.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Function.</a>

%F 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 a(n) ~ c * d^n, where d = -1/A143441 = 2.43161993449532399475429572773256778... and c = 0.765603960074106532799232452562411022387973764575133091283490410339311... - _Vaclav Kotesovec_, Jun 02 2018

%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma(k+1) * a(n-k). - _Ilya Gutkovskiy_, May 27 2020

%e 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 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 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / log( eta( -x + x^2 * O(x^n)))', n))} /* _Michael Somos_, Apr 05 2003 */

%Y Cf. A002040, A143348.

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_