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 A002040 Related to partitions. (Formerly M1159 N0442) 4
 1, 2, 4, 8, 21, 52, 131, 316, 765, 1846, 4494, 10944, 26654, 64798, 157502, 382868, 931028, 2264106, 5505777, 13387880, 32553601, 79156974, 192479838, 468039888, 1138098210, 2767421826, 6729311459, 16363118556, 39788886610, 96751470494 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 J. M. Gandhi, On numbers related to partitions of a number, Amer. Math. Monthly, 76 (1969), 1033-1036. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions. FORMULA 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 a(n) = sum_{k=0..n} (-1)^k*A000041(k)*A002039(n-k). - Mircea Merca, Feb 27 2014 a(n) ~ c * d^n, where d = -1/A143441 = 2.431619934495323994754... and c = 0.623278923942755977756856780504941340332933121682037117752100... - Vaclav Kotesovec, Jun 02 2018 EXAMPLE G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 21*x^4 + 52*x^5 + 131*x^6 + 316*x^7 + ... MATHEMATICA 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 *) 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 *) PROG (PARI) {a(n) = polcoeff( 1 / eta( -x + x^2 * O(x^n))', n)}; CROSSREFS Cf. A002039, A000203, A010815. Sequence in context: A059731 A001315 A216643 * A162110 A108071 A055876 Adjacent sequences:  A002037 A002038 A002039 * A002041 A002042 A002043 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Formula corrected and sequence extended by Michael Somos STATUS approved

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Last modified September 24 20:09 EDT 2018. Contains 315356 sequences. (Running on oeis4.)