OFFSET
0,4
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.
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
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
KEYWORD
nonn
AUTHOR
Joerg Arndt, Jan 04 2011
STATUS
approved