A179049 - OEIS

%I #64 Aug 06 2022 07:18:24

%S 1,1,0,2,0,2,1,3,1,3,3,4,4,4,6,6,8,6,12,8,14,10,19,13,23,16,29,21,35,

%T 26,43,34,50,43,61,54,72,67,85,84,100,103,119,126,138,155,163,186,191,

%U 224,224,268,263,319,308,378,360,447,422,523,494,614,576,716,674,833,787,964,917,1118

%N Odd-even partitions: number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is odd.

%C Parts are odd, even, odd, even, ...  [_Joerg Arndt_, Oct 27 2012]

%H Alois P. Heinz, <a href="/A179049/b179049.txt">Table of n, a(n) for n = 0..10000</a>

%H G. E. Andrews, <a href="https://doi.org/10.1016/0001-8708(84)90017-3">Ramanujan’s “lost” notebook. IV. Stacks and alternating parity in partitions</a>, Adv. in Math. 53 (1984), no. 1, 55-74.

%H Min-Joo Jang, <a href="https://arxiv.org/abs/1703.01837">Asymptotic behavior of odd-even partitions</a>, arXiv:1703.01837v1 [math.NT], 2017.

%F G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - x^(2*k)).

%F a(n) ~ (1/(2*sqrt(5)*n^(3/4)))*exp(Pi*sqrt(n/5)) [Jang 2017]. - _Peter Bala_, Mar 28 2017

%e From _Joerg Arndt_, Oct 27 2012:  (Start)

%e The a(20) = 14 such partitions of 20 are:

%e [ 1]  1 2 3 14

%e [ 2]  1 2 5 12

%e [ 3]  1 2 7 10

%e [ 4]  1 2 17

%e [ 5]  1 4 5 10

%e [ 6]  1 4 7 8

%e [ 7]  1 4 15

%e [ 8]  1 6 13

%e [ 9]  1 8 11

%e [10]  3 4 5 8

%e [11]  3 4 13

%e [12]  3 6 11

%e [13]  3 8 9

%e [14]  5 6 9

%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-> b(n, 1):

%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, Mod[t, 2], 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_ *)

%o (Sage)

%o odd_diffs = lambda x: all(abs(d) % 2 for d in differences(x))

%o satisfies = lambda p: not p or (min(p) % 2 and odd_diffs(p))

%o def A179049(n):

%o     return len([1 for p in Partitions(n,max_slope=-1) if satisfies(p)])

%o # _D. S. McNeil_, Jan 04 2011; adapted by _F. Chapoton_, Feb 24 2020

%o (Sage) # Alternative, after _Alois P. Heinz_:

%o def A179049(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)

%o [A179049(n) for n in range(70)] # _Peter Luschny_, Feb 25 2020

%o (PARI) N=99; x='x+O('x^N); Vec(sum(n=0,N, x^(n*(n+1)/2)/prod(k=1,n,1-x^(2*k))))

%Y Cf. A000009.

%Y Cf. A218355 (parts are even, odd, even, odd, ...).

%K nonn,easy

%O 0,4

%A _Joerg Arndt_, Jan 04 2011