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A179049 Odd-even partitions: number of partitions into distinct parts where all differences between consecutive parts are odd and the minimal part is odd. 5
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, 26, 43, 34, 50, 43, 61, 54, 72, 67, 85, 84, 100, 103, 119, 126, 138, 155, 163, 186, 191, 224, 224, 268, 263, 319, 308, 378, 360, 447, 422, 523, 494, 614, 576, 716, 674, 833, 787, 964, 917, 1118 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Parts are odd, even, odd, even, ...  [Joerg Arndt, Oct 27 2012]

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

G. E. Andrews, Ramanujan’s “lost” notebook. IV. Stacks and alternating parity in partitions, Adv. in Math. 53 (1984), no. 1, 55-74.

Min-Joo Jang, Asymptotic behavior of odd-even partitions, arXiv:1703.01837v1 [math.NT], 2017.

FORMULA

G.f.: sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n, 1-x^(2*k) ) ).

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

EXAMPLE

From Joerg Arndt, Oct 27 2012:  (Start)

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

[ 1]  1 2 3 14

[ 2]  1 2 5 12

[ 3]  1 2 7 10

[ 4]  1 2 17

[ 5]  1 4 5 10

[ 6]  1 4 7 8

[ 7]  1 4 15

[ 8]  1 6 13

[ 9]  1 8 11

[10]  3 4 5 8

[11]  3 4 13

[12]  3 6 11

[13]  3 8 9

[14]  5 6 9

(End)

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, irem(t, 2),

      `if`(i<1, 0, b(n, i-1, t)+`if`(i<=n and irem(i, 2)<>t,

       b(n-i, i-1, irem(i, 2)), 0)))

    end:

a:= n-> `if`(n=0, 1, add(b(n-i, i-1, irem(i, 2)), i=1..n)):

seq(a(n), n=0..100);  #  Alois P. Heinz, Nov 08 2012

MATHEMATICA

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 *)

PROG

(Sage)

def A179049(n):

....odd_diffs = lambda x: all(abs(d) % 2 == 1 for d in differences(x))

....satisfies = lambda p: not p or (min(p) % 2 == 1 and odd_diffs(p))

....return Partitions(n, max_slope=-1).filter(satisfies).cardinality()

# D. S. McNeil, Jan 04 2011_

(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))))

CROSSREFS

Cf. A000009.

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

Sequence in context: A025803 A029185 A029184 * A029221 A029183 A213423

Adjacent sequences:  A179046 A179047 A179048 * A179050 A179051 A179052

KEYWORD

nonn,easy

AUTHOR

Joerg Arndt, Jan 04 2011

STATUS

approved

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Last modified November 23 19:03 EST 2017. Contains 295128 sequences.