A002040 Related to partitions. (Formerly M1159 N0442)

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

0,2

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: A001315 A340595 A216643 * A162110 A108071 A337762

Adjacent sequences: A002037 A002038 A002039 * A002041 A002042 A002043

Keyword

nonn,easy,nice

Author

N. J. A. Sloane.

Extensions

Formula corrected and sequence extended by Michael Somos

Status

approved