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A179080
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Number of partitions of n into distinct parts where all differences between consecutive parts are odd.
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3
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n+1, 1-x^(2*k) ) ). - Joerg Arndt, Jan 29 2011
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EXAMPLE
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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)
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MAPLE
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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
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MATHEMATICA
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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 *)
Join[{1}, Table[Length[Select[IntegerPartitions[n], Max[Length/@Split[#]]==1 && AllTrue[ Differences[#], OddQ]&]], {n, 70}]] (* Harvey P. Dale, Jun 25 2022 *)
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PROG
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(Sage)
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
@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
(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 */
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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