A179080 Number of partitions of n into distinct parts where all differences between consecutive parts are odd.

1, 1, 1, 2, 1, 3, 2, 4, 2, 6, 4, 7, 5, 9, 8, 12, 10, 14, 15, 17, 19, 22, 26, 26, 32, 32, 42, 40, 52, 48, 66, 59, 79, 73, 98, 89, 118, 108, 143, 133, 170, 160, 204, 194, 241, 236, 286, 283, 336, 339, 396, 407, 464, 483, 544, 575, 634, 681, 740, 803, 862, 944, 1001, 1110, 1162, 1296, 1348, 1512, 1561, 1760, 1805

Offset: 0

Author

Joerg Arndt, Jan 04 2011

Keywords

• nonn

Examples

			From _Joerg Arndt_, Oct 27 2012:  (Start)
The a(18) = 15 such partitions of 18 are:
[ 1]  1 2 3 12
[ 2]  1 2 5 10
[ 3]  1 2 7 8
[ 4]  1 2 15
[ 5]  1 4 5 8
[ 6]  1 4 13
[ 7]  1 6 11
[ 8]  1 8 9
[ 9]  2 3 4 9
[10]  2 3 6 7
[11]  3 4 5 6
[12]  3 4 11
[13]  3 6 9
[14]  5 6 7
[15]  18
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Atul Dixit and Gaurav Kumar, The Rogers-Ramanujan dissection of a theta function, arXiv:2411.06412 [math.NT], 2024. See pp. 16, 23.

Crossrefs

Cf. A179049 (odd differences and odd minimal part).

Cf. A189357 (even differences, distinct parts), A096441 (even differences).

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

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-> `if`(n=0, 1, b(n, 1)+b(n, 2)):
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, 1, 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 *)
Join[{1},Table[Length[Select[IntegerPartitions[n],Max[Length/@Split[#]]==1 && AllTrue[ Differences[#],OddQ]&]],{n,70}]] (* Harvey P. Dale, Jun 25 2022 *)

PARI

N=66; x='x+O('x^N); gf = sum(n=0,N, x^(n*(n+1)/2) / prod(k=1,n+1, 1-x^(2*k) ) ); Vec( gf ) /* Joerg Arndt, Jan 29 2011 */

Sage

def A179080(n):
    odd_diffs = lambda x: all(abs(d) % 2 == 1 for d in differences(x))
    satisfies = lambda p: not p or odd_diffs(p)
    def count(pred, iter): return sum(1 for item in iter if pred(item))
    return count(satisfies, Partitions(n, max_slope=-1))
print([A179080(n) for n in range(0, 20)]) # show first terms

Sage

# Alternative after Alois P. Heinz:
def A179080(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) + h(n, 2) if n > 0 else 1
print([A179080(n) for n in range(71)]) # Peter Luschny, Feb 25 2020

Formula

G.f.: sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n+1, 1-x^(2*k) ) ). - Joerg Arndt, Jan 29 2011

a(n) = A179049(n) + A218355(n). - Joerg Arndt, Oct 27 2012