A189357 Number of partitions of n into distinct parts where all differences between consecutive parts are even.

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 2, 7, 3, 8, 4, 11, 5, 13, 6, 17, 8, 20, 9, 26, 12, 30, 14, 38, 17, 45, 20, 55, 25, 64, 29, 79, 35, 91, 41, 110, 49, 128, 57, 152, 68, 176, 78, 209, 93, 240, 107, 282, 125, 325, 144, 379, 168, 434, 192, 505, 223, 576, 255, 666, 294, 760, 335, 873, 385, 993, 437, 1139

Offset: 0

Author

Joerg Arndt, Apr 20 2011

Keywords

• nonn

Comments

Also number of partitions into distinct parts where either all parts are even or all parts are odd.

Also number of symmetric unimodal compositions of n where the maximal part m appears at least m times, see example. [Joerg Arndt, Jun 11 2013]

Examples

			a(14)=8 because there are 8 such partitions of 14: 1+13 =2+4+8 =2+12 =3+11 =4+10 =5+9 =6+8 =14
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 +...
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(18)=13 symmetric unimodal compositions of 18 where the maximal part m appears at least m times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 ]
03:  [ 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 ]
04:  [ 1 1 1 1 1 2 2 2 2 1 1 1 1 1 ]
05:  [ 1 1 1 1 2 2 2 2 2 1 1 1 1 ]
06:  [ 1 1 1 2 2 2 2 2 2 1 1 1 ]
07:  [ 1 1 1 3 3 3 3 1 1 1 ]
08:  [ 1 1 2 2 2 2 2 2 2 1 1 ]
09:  [ 1 2 2 2 2 2 2 2 2 1 ]
10:  [ 1 2 3 3 3 3 2 1 ]
11:  [ 1 4 4 4 4 1 ]
12:  [ 2 2 2 2 2 2 2 2 2 ]
13:  [ 3 3 3 3 3 3 ]
(End)
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009 (partitions of 2*n with even differences and even minimal part), A179080 (odd differences), A179049 (odd differences and odd minimal part).

Programs

PARI

{a(n)=polcoeff(-1 + prod(m=1,n\1, 1+x^(2*m))+prod(m=1,n\2+1,1+x^(2*m-1))+x*O(x^n),n)} /* Paul D. Hanna */

PARI

{a(n)=polcoeff(-1 + sum(m=0,sqrtint(n+1), x^(m^2)*(1+x^m)/prod(k=1,m,1-x^(2*k)+x*O(x^n))),n)} /* Paul D. Hanna */

Sage

def A189357(n):
    works = lambda part: all(x % 2 == 0 for x in differences(part))
    def count(pred, iter): return sum(1 for item in iter if pred(item))
    return count(works, Partitions(n, max_slope=-1))
print([A189357(n) for n in range(0, 30)])
# D. S. McNeil, Apr 21 2011 (updated to Python3 by Peter Luschny, Mar 06 2020 )

Formula

a(n) = A000009(2*n) + A000700(n) for n>=1, a(0)=1.

G.f.: -1 + prod(n>=1, 1+x^(2*n) ) + prod(n>=1, 1+x^(2*n-1) ).

G.f.: -1 + sum(n>=0, x^(n^2)*(1+x^n) / prod(k=1..n, 1-x^(2*k)) ). [Joerg Arndt, Jan 27 2011]