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A002039 Convolution inverse of A143348.
(Formerly M2465 N0979)
5
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-Fran├žois 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,changed

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified May 28 01:48 EDT 2020. Contains 334671 sequences. (Running on oeis4.)