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A002040 Related to partitions.

%I A002040 M1159 N0442 #44 Jul 08 2025 16:27:32
%S A002040 1,2,4,8,21,52,131,316,765,1846,4494,10944,26654,64798,157502,382868,
%T A002040 931028,2264106,5505777,13387880,32553601,79156974,192479838,
%U A002040 468039888,1138098210,2767421826,6729311459,16363118556,39788886610,96751470494
%N A002040 Related to partitions.
%D A002040 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002040 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002040 Vincenzo Librandi, <a href="/A002040/b002040.txt">Table of n, a(n) for n = 0..300</a>
%H A002040 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 A002040 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions.</a>
%F A002040 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 A002040 a(n) = sum_{k=0..n} (-1)^k*A000041(k)*A002039(n-k). - _Mircea Merca_, Feb 27 2014
%F A002040 a(n) ~ c * d^n, where d = -1/A143441 = 2.431619934495323994754... and c = 0.623278923942755977756856780504941340332933121682037117752100... - _Vaclav Kotesovec_, Jun 02 2018
%e A002040 G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 21*x^4 + 52*x^5 + 131*x^6 + 316*x^7 + ...
%t A002040 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 A002040 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 A002040 (PARI) {a(n) = polcoeff( 1 / eta( -x + x^2 * O(x^n))', n)};
%Y A002040 Cf. A002039, A000203, A010815.
%K A002040 nonn,easy,nice
%O A002040 0,2
%A A002040 _N. J. A. Sloane_
%E A002040 Formula corrected and sequence extended by _Michael Somos_