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

%I A179080 #54 Nov 18 2024 16:14:24
%S A179080 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 A179080 52,48,66,59,79,73,98,89,118,108,143,133,170,160,204,194,241,236,286,
%U A179080 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 A179080 Number of partitions of n into distinct parts where all differences between consecutive parts are odd.
%H A179080 Alois P. Heinz, <a href="/A179080/b179080.txt">Table of n, a(n) for n = 0..10000</a>
%H A179080 Atul Dixit and Gaurav Kumar, <a href="https://arxiv.org/abs/2411.06412">The Rogers-Ramanujan dissection of a theta function</a>, arXiv:2411.06412 [math.NT], 2024. See pp. 16, 23.
%F A179080 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 A179080 a(n) = A179049(n) + A218355(n). - _Joerg Arndt_, Oct 27 2012
%e A179080 From _Joerg Arndt_, Oct 27 2012:  (Start)
%e A179080 The a(18) = 15 such partitions of 18 are:
%e A179080 [ 1]  1 2 3 12
%e A179080 [ 2]  1 2 5 10
%e A179080 [ 3]  1 2 7 8
%e A179080 [ 4]  1 2 15
%e A179080 [ 5]  1 4 5 8
%e A179080 [ 6]  1 4 13
%e A179080 [ 7]  1 6 11
%e A179080 [ 8]  1 8 9
%e A179080 [ 9]  2 3 4 9
%e A179080 [10]  2 3 6 7
%e A179080 [11]  3 4 5 6
%e A179080 [12]  3 4 11
%e A179080 [13]  3 6 9
%e A179080 [14]  5 6 7
%e A179080 [15]  18
%e A179080 (End)
%p A179080 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A179080       `if`(i>n, 0, b(n, i+2)+b(n-i, i+1)))
%p A179080     end:
%p A179080 a:= n-> `if`(n=0, 1, b(n, 1)+b(n, 2)):
%p A179080 seq(a(n), n=0..100);  # _Alois P. Heinz_, Nov 08 2012; revised Feb 24 2020
%t A179080 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 A179080 Join[{1},Table[Length[Select[IntegerPartitions[n],Max[Length/@Split[#]]==1 && AllTrue[ Differences[#],OddQ]&]],{n,70}]] (* _Harvey P. Dale_, Jun 25 2022 *)
%o A179080 (Sage)
%o A179080 def A179080(n):
%o A179080     odd_diffs = lambda x: all(abs(d) % 2 == 1 for d in differences(x))
%o A179080     satisfies = lambda p: not p or odd_diffs(p)
%o A179080     def count(pred, iter): return sum(1 for item in iter if pred(item))
%o A179080     return count(satisfies, Partitions(n, max_slope=-1))
%o A179080 print([A179080(n) for n in range(0, 20)]) # show first terms
%o A179080 (Sage) # Alternative after _Alois P. Heinz_:
%o A179080 def A179080(n):
%o A179080     @cached_function
%o A179080     def h(n, k):
%o A179080         if n == 0: return 1
%o A179080         if k  > n: return 0
%o A179080         return h(n, k+2) + h(n-k, k+1)
%o A179080     return h(n, 1) + h(n, 2) if n > 0 else 1
%o A179080 print([A179080(n) for n in range(71)]) # _Peter Luschny_, Feb 25 2020
%o A179080 (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 A179080 Cf. A179049 (odd differences and odd minimal part).
%Y A179080 Cf. A189357 (even differences, distinct parts), A096441 (even differences).
%Y A179080 Cf. A000009 (partitions of 2*n with even differences and even minimal part).
%K A179080 nonn
%O A179080 0,4
%A A179080 _Joerg Arndt_, Jan 04 2011