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A332305
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Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.
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11
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1, 0, 0, 2, 2, 4, 4, 6, 6, 8, 32, 34, 58, 84, 132, 158, 230, 280, 376, 450, 570, 1388, 1556, 2398, 3310, 4920, 6600, 9674, 12122, 16684, 21340, 28110, 34974, 45392, 55208, 69274, 124498, 143676, 204012, 270758, 377966, 493024, 690304, 895434, 1223826, 1562948
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OFFSET
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0,4
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions
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FORMULA
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G.f.: Sum_{k>=0} (2*k)! * x^(k*(2*k + 1)) / Product_{j=1..2*k} (1 - x^j).
a(n) = A032020(n) - A332304(n).
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EXAMPLE
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a(5) = 4 because we have [4, 1], [3, 2], [2, 3] and [1, 4].
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MAPLE
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b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
irem(p+1, 2)*p!, add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55); # Alois P. Heinz, Feb 09 2020
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MATHEMATICA
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nmax = 45; CoefficientList[Series[Sum[(2 k)! x^(k (2 k + 1))/Product[1 - x^j, {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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Cf. A027187, A032020, A034008, A067661, A332304.
Sequence in context: A259881 A238132 A278296 * A340282 A008642 A001364
Adjacent sequences: A332302 A332303 A332304 * A332306 A332307 A332308
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, Feb 09 2020
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STATUS
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approved
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