login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A338860 The excess of the number of partitions of n with more odd parts than even parts over the number of partitions of n with more even parts than odd parts. 1
0, 1, 0, 2, 1, 3, 4, 6, 8, 11, 17, 21, 30, 38, 53, 68, 90, 115, 150, 192, 243, 312, 390, 496, 613, 775, 951, 1193, 1456, 1810, 2200, 2715, 3285, 4026, 4856, 5909, 7106, 8595, 10301, 12394, 14809, 17728, 21118, 25171, 29891, 35489, 42018, 49702, 58678, 69180 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2000

B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

FORMULA

G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2-n)*(1-q^n)/Product_{k=1..n} (1-q^(2*k))^2.

a(n) = A108950(n) - A108949(n).

EXAMPLE

The 3 partitions of 4 with more odd parts than even parts are [3,1], [2,1,1], and [1,1,1,1], while the 2 partitions of 4 with more even parts than odd parts are [4] and [2,2].   Hence a(4) = 3-2 = 1.

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, signum(t), `if`(i<1, 0,

      b(n, i-1, t)+ b(n-i, min(n-i, i), t+(2*irem(i, 2)-1))))

    end:

a:= n-> b(n$2, 0):

seq(a(n), n=0..55);  # Alois P. Heinz, Jan 14 2021

PROG

(PARI) for(n=0, 43, my(me=0, mo=0); forpart(v=n, my(x=Vec(v), se=sum(k=1, #x, x[k]%2==0), so=sum(k=1, #x, x[k]%2>0)); me+=(se>so); mo+=(so>se)); print1(mo-me, ", ")) \\ Hugo Pfoertner, Jan 13 2021

CROSSREFS

Cf. A108949, A108950.

Sequence in context: A293253 A266687 A325271 * A060214 A259773 A030133

Adjacent sequences:  A338857 A338858 A338859 * A338861 A338862 A338863

KEYWORD

nonn

AUTHOR

Jeremy Lovejoy, Jan 12 2021

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 20 18:45 EDT 2021. Contains 343137 sequences. (Running on oeis4.)