OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Feb 16 2014
a(n) is the number of partitions of 3n-2 having n as a part, for n >= 1. Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1. - Clark Kimberling, Mar 02 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(x^3, x^5) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 16 sequence [ 2, 2, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, ...]. - Michael Somos, Apr 25 2003
a(n) = A000041(2*n).
G.f.: Product_{k>=1} (1 + x^(8*k-4)) * (1 + x^(8*k)) * (1 + x^k)^2 / ((1 + x^(8*k-1)) * (1 + x^(8*k-7)) * (1 - x^k)). - Vaclav Kotesovec, Nov 17 2016
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Feb 16 2022
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 11*x^3 + 22*x^4 + 42*x^5 + 77*x^6 + 135*x^7 + ...
G.f. = q^-1 + 2*q^47 + 5*q^95 + 11*q^143 + 22*q^191 + 42*q^239 + 77*q^287 + ...
MAPLE
a:= n-> combinat[numbpart](2*n):
seq(a(n), n=0..42); # Alois P. Heinz, Jan 29 2020
MATHEMATICA
nn=100; Table[CoefficientList[Series[Product[1/(1-x^i), {i, 1, nn}], {x, 0, nn}], x][[2i-1]], {i, 1, nn/2}] (* Geoffrey Critzer, Sep 28 2013 *)
(* also *)
Table[PartitionsP[2 n], {n, 0, 40}] (* Clark Kimberling, Mar 02 2014 *)
(* also *)
Table[Count[IntegerPartitions[3 n - 2], p_ /; MemberQ[p, n]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(8*k-4))*(1 + x^(8*k))*(1 + x^k)^2/((1 + x^(8*k-1))*(1 + x^(8*k-7))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 1))), 2*n))}; /* Michael Somos, Apr 25 2003 */
(PARI) a(n) = numbpart(2*n); \\ Michel Marcus, Sep 28 2013
(MuPAD) combinat::partitions::count(2*i) $i=0..54 // Zerinvary Lajos, Apr 16 2007
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 31 2000
STATUS
approved