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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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

Example

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)

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 *)

Prog

(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
(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 */

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).

Sequence in context: A024162 A334677 A365876 * A294199 A078658 A307719

Adjacent sequences: A179077 A179078 A179079 * A179081 A179082 A179083

Keyword

nonn

Author

Joerg Arndt, Jan 04 2011

Status

approved