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A179080 Number of partitions of n into distinct parts where all differences between consecutive parts are odd. 3
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, 52, 48, 66, 59, 79, 73, 98, 89, 118, 108, 143, 133, 170, 160, 204, 194, 241, 236, 286, 283, 336, 339, 396, 407, 464, 483, 544, 575, 634, 681, 740, 803, 862, 944, 1001, 1110, 1162, 1296, 1348, 1512, 1561, 1760, 1805 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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

a(n) = A179049(n) + A218355(n). - Joerg Arndt, Oct 27 2012

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-Fran├žois Alcover, Mar 24 2015, after Alois P. Heinz *)

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

Cf. A179049 (odd differences and odd minimal part).

Cf. A189357 (even differences, distinct parts), A096441 (even differences).

Cf. A000009 (partitions of 2*n with even differences and even minimal part).

Sequence in context: A308308 A024162 A334677 * A294199 A078658 A307719

Adjacent sequences:  A179077 A179078 A179079 * A179081 A179082 A179083

KEYWORD

nonn

AUTHOR

Joerg Arndt, Jan 04 2011

STATUS

approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)