A002040 Related to partitions.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 21*x^4 + 52*x^5 + 131*x^6 + 316*x^7 + ...
		

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.

Crossrefs

Cf. A002039, A000203, A010815.

Programs

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 *)

PARI

{a(n) = polcoeff( 1 / eta( -x + x^2 * O(x^n))', n)};

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

Extensions

Formula corrected and sequence extended by Michael Somos