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A087897
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Number of partitions of n into odd parts greater than 1.
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34
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1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756
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OFFSET
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0,10
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COMMENTS
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Also number of partitions of n into distinct parts which are not powers of 2.
Also number of partitions of n into distinct parts such that the two largest parts differ by 1.
Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006
Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007
In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).
Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013
Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014
Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:
32 222 322 332 432 3322 3332 4332 4432 5432 43332
2222 3222 22222 4322 33222 33322 33332 44322
32222 222222 43222 43322 333222
322222 332222 432222
2222222 3222222
(End)
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REFERENCES
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J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I
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LINKS
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Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160.
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FORMULA
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Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q.
Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function.
G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012
G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)).
G.f.: Product_{k >= 1, k not a power of 2} (1+x^k).
G.f.: Sum_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - Emeric Deutsch, Mar 30 2006
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016
G.f.: 1/(1 - x^3) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 - x^3)*(1 - x^5)) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = ..., extending Deutsch's result dated Mar 30 2006. - Peter Bala, Jan 15 2021
G.f.: Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 2..2*n+1} (1 - x^k). (Set z = x^3 and q = x^2 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
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EXAMPLE
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1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ...
q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ...
a(10)=2 because we have [7,3] and [5,5].
There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times:
01: [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]
03: [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]
04: [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]
05: [ 1 1 1 2 5 5 5 2 1 1 1 ]
06: [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]
07: [ 1 1 3 5 5 5 3 1 1 ]
08: [ 1 1 7 7 7 1 1 ]
09: [ 1 2 2 5 5 5 2 2 1 ]
10: [ 1 4 5 5 5 4 1 ]
11: [ 2 2 2 2 3 3 3 2 2 2 2 ]
12: [ 2 3 5 5 5 3 2 ]
13: [ 2 7 7 7 2 ]
(End)
The a(7) = 1 through a(19) = 8 partitions are the following (A..J = 10..19). The Heinz numbers of these partitions are given by A341449.
7 53 9 55 B 75 D 77 F 97 H 99 J
333 73 533 93 553 95 555 B5 755 B7 775
3333 733 B3 753 D3 773 D5 955
5333 933 5533 953 F3 973
33333 7333 B33 5553 B53
53333 7533 D33
9333 55333
333333 73333
(End)
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MAPLE
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To get 128 terms: t4 := mul((1+x^(2^n)), n=0..7); t5 := mul((1+x^k), k=1..128): t6 := series(t5/t4, x, 100); t7 := seriestolist(t6);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
end:
a:= n-> b(n, n-1+irem(n, 2)):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *)
Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&OddQ[Times@@#]&]], {n, 0, 30}] (* Gus Wiseman, Feb 16 2021 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */
(Haskell)
a087897 = p [3, 5..] where
p [] _ = 0
p _ 0 = 1
p ks'@(k:ks) m | m < k = 0
| otherwise = p ks' (m - k) + p ks m
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
if n==0: return 1
if k<3 or n<0: return 0
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CROSSREFS
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The version for factorizations is A340101.
Partitions whose only even part is the smallest are counted by A341447.
The Heinz numbers of these partitions are given by A341449.
A025147 counts strict partitions with no 1's.
A025148 counts strict partitions with no 1's or 2's.
A026804 counts partitions whose smallest part is odd, ranked by A340932.
A340385 counts partitions with odd length and maximum, ranked by A340386.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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