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

0,5

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]

Links

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

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]

Example

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)

Prog

(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 )
(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 */

Crossrefs

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

Sequence in context: A323523 A371092 A124072 * A100053 A029194 A246582

Adjacent sequences: A189354 A189355 A189356 * A189358 A189359 A189360

Keyword

nonn

Author

Joerg Arndt, Apr 20 2011

Status

approved