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%I #50 Jun 25 2022 15:29:31
%S 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,
%T 52,48,66,59,79,73,98,89,118,108,143,133,170,160,204,194,241,236,286,
%U 283,336,339,396,407,464,483,544,575,634,681,740,803,862,944,1001,1110,1162,1296,1348,1512,1561,1760,1805
%N Number of partitions of n into distinct parts where all differences between consecutive parts are odd.
%H Alois P. Heinz, <a href="/A179080/b179080.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n+1, 1-x^(2*k) ) ). - _Joerg Arndt_, Jan 29 2011
%F a(n) = A179049(n) + A218355(n). - _Joerg Arndt_, Oct 27 2012
%e From _Joerg Arndt_, Oct 27 2012: (Start)
%e The a(18) = 15 such partitions of 18 are:
%e [ 1] 1 2 3 12
%e [ 2] 1 2 5 10
%e [ 3] 1 2 7 8
%e [ 4] 1 2 15
%e [ 5] 1 4 5 8
%e [ 6] 1 4 13
%e [ 7] 1 6 11
%e [ 8] 1 8 9
%e [ 9] 2 3 4 9
%e [10] 2 3 6 7
%e [11] 3 4 5 6
%e [12] 3 4 11
%e [13] 3 6 9
%e [14] 5 6 7
%e [15] 18
%e (End)
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
%p end:
%p a:= n-> `if`(n=0, 1, b(n, 1)+b(n, 2)):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Nov 08 2012; revised Feb 24 2020
%t 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_ *)
%t Join[{1},Table[Length[Select[IntegerPartitions[n],Max[Length/@Split[#]]==1 && AllTrue[ Differences[#],OddQ]&]],{n,70}]] (* _Harvey P. Dale_, Jun 25 2022 *)
%o (Sage)
%o def A179080(n):
%o odd_diffs = lambda x: all(abs(d) % 2 == 1 for d in differences(x))
%o satisfies = lambda p: not p or odd_diffs(p)
%o def count(pred, iter): return sum(1 for item in iter if pred(item))
%o return count(satisfies, Partitions(n, max_slope=-1))
%o print([A179080(n) for n in range(0, 20)]) # show first terms
%o (Sage) # Alternative after _Alois P. Heinz_:
%o def A179080(n):
%o @cached_function
%o def h(n, k):
%o if n == 0: return 1
%o if k > n: return 0
%o return h(n, k+2) + h(n-k, k+1)
%o return h(n, 1) + h(n, 2) if n > 0 else 1
%o print([A179080(n) for n in range(71)]) # _Peter Luschny_, Feb 25 2020
%o (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 */
%Y Cf. A179049 (odd differences and odd minimal part).
%Y Cf. A189357 (even differences, distinct parts), A096441 (even differences).
%Y Cf. A000009 (partitions of 2*n with even differences and even minimal part).
%K nonn
%O 0,4
%A _Joerg Arndt_, Jan 04 2011
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