A179049 Odd-even partitions: number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is odd.

1, 1, 0, 2, 0, 2, 1, 3, 1, 3, 3, 4, 4, 4, 6, 6, 8, 6, 12, 8, 14, 10, 19, 13, 23, 16, 29, 21, 35, 26, 43, 34, 50, 43, 61, 54, 72, 67, 85, 84, 100, 103, 119, 126, 138, 155, 163, 186, 191, 224, 224, 268, 263, 319, 308, 378, 360, 447, 422, 523, 494, 614, 576, 716, 674, 833, 787, 964, 917, 1118

Offset: 0

Author

Joerg Arndt, Jan 04 2011

Keywords

• nonn
• easy

Comments

Parts are odd, even, odd, even, ... [Joerg Arndt, Oct 27 2012]

Examples

			From _Joerg Arndt_, Oct 27 2012:  (Start)
The a(20) = 14 such partitions of 20 are:
[ 1]  1 2 3 14
[ 2]  1 2 5 12
[ 3]  1 2 7 10
[ 4]  1 2 17
[ 5]  1 4 5 10
[ 6]  1 4 7 8
[ 7]  1 4 15
[ 8]  1 6 13
[ 9]  1 8 11
[10]  3 4 5 8
[11]  3 4 13
[12]  3 6 11
[13]  3 8 9
[14]  5 6 9
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• G. E. Andrews, Ramanujan’s “lost” notebook. IV. Stacks and alternating parity in partitions, Adv. in Math. 53 (1984), no. 1, 55-74.
• Min-Joo Jang, Asymptotic behavior of odd-even partitions, arXiv:1703.01837v1 [math.NT], 2017.

Crossrefs

Cf. A000009.

Cf. A218355 (parts are even, odd, even, odd, ...).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 08 2012; revised Feb 24 2020

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, Mod[t, 2], If[i<1, 0, b[n, i-1, t] + If[i <= n && Mod[i, 2] != t, b[n-i, i-1, Mod[i, 2]], 0]]]; a[n_] := If[n==0, 1, Sum[b[n-i, i-1, Mod[i, 2]], {i, 1, n}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

PARI

N=99; x='x+O('x^N); Vec(sum(n=0,N, x^(n*(n+1)/2)/prod(k=1,n,1-x^(2*k))))

Sage

odd_diffs = lambda x: all(abs(d) % 2 for d in differences(x))
satisfies = lambda p: not p or (min(p) % 2 and odd_diffs(p))
def A179049(n):
    return len([1 for p in Partitions(n,max_slope=-1) if satisfies(p)])
# D. S. McNeil, Jan 04 2011; adapted by F. Chapoton, Feb 24 2020

Sage

# Alternative, after Alois P. Heinz:
def A179049(n):
    @cached_function
    def h(n, k):
        if n == 0: return 1
        if k  > n: return 0
        return h(n, k+2) + h(n-k, k+1)
    return h(n, 1)
[A179049(n) for n in range(70)] # Peter Luschny, Feb 25 2020

Formula

G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - x^(2*k)).

a(n) ~ (1/(2*sqrt(5)*n^(3/4)))*exp(Pi*sqrt(n/5)) [Jang 2017]. - Peter Bala, Mar 28 2017