|
|
A038348
|
|
Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).
|
|
17
|
|
|
1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003
Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003
Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005
Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
|
|
EXAMPLE
|
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (31) (32) (42) (43)
(111) (211) (41) (51) (52)
(1111) (311) (321) (61)
(2111) (411) (421)
(11111) (3111) (511)
(21111) (3211)
(111111) (4111)
(31111)
(211111)
(1111111)
(End)
|
|
MAPLE
|
f:=1/(1-x^2)/product(1-x^(2*j-1), j=1..32): fser:=series(f, x=0, 62): seq(coeff(fser, x, n), n=0..58); # Emeric Deutsch, Feb 22 2006
|
|
MATHEMATICA
|
mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)
|
|
PROG
|
(SageMath) # uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|