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 A002039 Convolution inverse of A143348. (Formerly M2465 N0979) 7
 1, 3, 5, 10, 25, 64, 160, 390, 940, 2270, 5515, 13440, 32735, 79610, 193480, 470306, 1143585, 2781070, 6762990, 16445100, 39987325, 97232450, 236432060, 574915770, 1397981470, 3399360474, 8265943685, 20099618590, 48874630750 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Gandhi denotes f(-x) by Phi(x) and a(n) by G(n). 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..1000 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 Function. FORMULA 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 a(n) ~ c * d^n, where d = -1/A143441 = 2.43161993449532399475429572773256778... and c = 0.765603960074106532799232452562411022387973764575133091283490410339311... - Vaclav Kotesovec, Jun 02 2018 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma(k+1) * a(n-k). - Ilya Gutkovskiy, May 27 2020 EXAMPLE 1 + 3*x + 5*x^2 + 10*x^3 + 25*x^4 + 64*x^5 + 160*x^6 + 390*x^7 + 940*x^8 + ... MATHEMATICA 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 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / log( eta( -x + x^2 * O(x^n)))', n))} /* Michael Somos, Apr 05 2003 */ CROSSREFS Cf. A002040, A143348. Sequence in context: A240619 A171867 A243513 * A243514 A243515 A243516 Adjacent sequences: A002036 A002037 A002038 * A002040 A002041 A002042 KEYWORD nonn,nice,easy AUTHOR N. J. A. Sloane, Simon Plouffe STATUS approved

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Last modified July 16 10:51 EDT 2024. Contains 374345 sequences. (Running on oeis4.)