OFFSET
1,2
COMMENTS
Note that this is very similar to the "crank" of Andrews and Garvan. The number of partitions pi with P(pi) odd is the given sequence.
The sequence is the same as A087787 except for the value of a(1) (this was established by George Andrews, Jan 18 2005). If "even" is replace by "odd" in the definition of the sequence, the new sequence is almost identical except for two values and a shift to the right.
Also, positive integers of A182712. a(n) is also the number of 2´s in the n-th row that contain a 2 as a part in the triangle of A138121 (note that rows 1 and 3 do not contain a 2 as a part). - Omar E. Pol, Nov 28 2010
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
FORMULA
G.f.: x+(1/(1+x))* Product_{n>=1}(1/(1-x^n)). [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = A000041(n) - a(n-1), for n>2. - Jon Maiga, Aug 29 2019 [corrected by Vaclav Kotesovec, Aug 29 2019]
EXAMPLE
a(3)=1 because P(3)=3, P(2 1)=1 and P(1 1 1)=0.
MATHEMATICA
Rest[ CoefficientList[ Series[x + 1/(1 + x) Product[1/(1 - x^n), {n, 50}], {x, 0, 50}], x]] (* Robert G. Wilson v, Feb 11 2005 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, Jan 13 2005
EXTENSIONS
More terms from Robert G. Wilson v, Feb 11 2005
STATUS
approved