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

0,4

Comments

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

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.

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

Example

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)

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 *)

Prog

(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
(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))))

Crossrefs

Cf. A000009.

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

Sequence in context: A025803 A029185 A029184 * A341976 A029221 A304034

Adjacent sequences: A179046 A179047 A179048 * A179050 A179051 A179052

Keyword

nonn,easy

Author

Joerg Arndt, Jan 04 2011

Status

approved