%I M1159 N0442 #39 Jul 12 2018 20:27:24

%S 1,2,4,8,21,52,131,316,765,1846,4494,10944,26654,64798,157502,382868,

%T 931028,2264106,5505777,13387880,32553601,79156974,192479838,

%U 468039888,1138098210,2767421826,6729311459,16363118556,39788886610,96751470494

%N Related to partitions.

%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="/A002040/b002040.txt">Table of n, a(n) for n = 0..300</a>

%H J. M. Gandhi, <a href="https://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 Functions.</a>

%F G.f.: 1/(f(q)') where f(-q)=Product_{k>0} (1-q^k) is one of Ramanujan's theta functions. - _Michael Somos_, Apr 08 2003

%F a(n) = sum_{k=0..n} (-1)^k*A000041(k)*A002039(n-k). - _Mircea Merca_, Feb 27 2014

%F a(n) ~ c * d^n, where d = -1/A143441 = 2.431619934495323994754... and c = 0.623278923942755977756856780504941340332933121682037117752100... - _Vaclav Kotesovec_, Jun 02 2018

%e G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 21*x^4 + 52*x^5 + 131*x^6 + 316*x^7 + ...

%t max = 29; f[q_] := Product[1 - (-q)^k, {k, 1, max + 1}]; CoefficientList[ Series[1/f'[q], {q, 0, max}], q] (* _Jean-François Alcover_, Jun 18 2012, after _Michael Somos_ *)

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 1 / D[ Normal @ Series[ QPochhammer[ -x], {x, 0, n + 1}], x], {x, 0, n}]]; (* _Michael Somos_, May 31 2016 *)

%o (PARI) {a(n) = polcoeff( 1 / eta( -x + x^2 * O(x^n))', n)};

%Y Cf. A002039, A000203, A010815.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_.

%E Formula corrected and sequence extended by _Michael Somos_