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A101707
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Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).
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21
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0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750
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OFFSET
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0,5
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COMMENTS
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REFERENCES
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George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
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LINKS
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FORMULA
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G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
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EXAMPLE
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a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
11 . 31 32 33 52 53 54 55
1111 51 3211 71 72 73
3111 3221 3222 91
111111 3311 3321 3322
5111 5211 3331
311111 321111 5221
11111111 5311
7111
322111
331111
511111
31111111
1111111111
Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
2 . 4 221 6 421 8 432 A
211 222 22111 422 441 433
411 431 621 442
21111 611 22221 622
22211 42111 631
41111 2211111 811
2111111 22222
42211
43111
61111
2221111
4111111
211111111
(End)
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MAPLE
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b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
`if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
`if`(r<0, irem(i, 2), r))))
end:
a:= n-> b(n$2, -1)/2:
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&OddQ[Max[#]]&]], {n, 0, 30}] (* Gus Wiseman, Feb 10 2021 *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
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CROSSREFS
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Note: A-numbers of ranking sequences are in parentheses below.
The even- but not necessarily positive-rank version is A340601 (A340602).
The Heinz numbers of these partitions are (A340604).
- Rank -
A257541 gives the rank of the partition with Heinz number n.
- Odd -
A026804 counts partitions whose least part is odd.
A339890 counts factorizations of odd length.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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