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A067659
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Number of partitions of n into distinct parts such that number of parts is odd.
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85
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0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404
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OFFSET
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0,7
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COMMENTS
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Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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LINKS
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FORMULA
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Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006
G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [Joerg Arndt, Apr 01 2014]
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EXAMPLE
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The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
5 6 7 8 9 A B C D E F
321 421 431 432 532 542 543 643 653 654
521 531 541 632 642 652 743 753
621 631 641 651 742 752 762
721 731 732 751 761 843
821 741 832 842 852
831 841 851 861
921 931 932 942
A21 941 951
A31 A32
B21 A41
B31
C21
54321
(End)
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= n-> b(n$2, 0):
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&]], {n, 0, 30}] (* Gus Wiseman, Jan 09 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( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */
(PARI) N=66; q='q+O('q^N); S=1+2*sqrtint(N);
gf=sum(n=1, S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );
(PARI) N=66; q='q+O('q^N); S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
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CROSSREFS
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Numbers with these strict partitions as binary indices are A000069.
The Heinz numbers of these partitions are A030059.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A174726 counts ordered factorizations of odd length.
- A339890 counts factorizations of odd length.
A008289 counts strict partitions by sum and length.
A026804 counts partitions whose least part is odd, with strict case A026832.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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