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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
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
Sequence in context: A240619 A171867 A243513 * A243514 A243515 A243516
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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