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A324595
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Number of colored integer partitions of 2n such that all colors from an n-set are used and parts differ by size or by color.
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7
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1, 1, 5, 19, 85, 381, 1751, 8135, 38173, 180415, 857695, 4096830, 19645975, 94523729, 456079769, 2206005414, 10693086637, 51930129399, 252617434619, 1230714593340, 6003931991895, 29325290391416, 143393190367102, 701862880794183, 3438561265961263
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listen;
history;
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OFFSET
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0,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1433
Wikipedia, Partition (number theory)
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FORMULA
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a(n) = A308680(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.0032778445310926321307990027... and c = 0.2798596129161126875318997... - Vaclav Kotesovec, Sep 14 2019
a(n) = [x^(2n)] (-1 + Product_{j>=1} (1 + x^j))^n. - Alois P. Heinz, Jan 29 2021
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EXAMPLE
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a(2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
end:
a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n+1),
(q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> g(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Jan 29 2021
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k] Binomial[k, j]][n - i j], {j, 0, Min[k, n/i]}]]];
a[n_] := Sum[b[2n, 2n, n - i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, May 06 2020, after Maple *)
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CROSSREFS
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Cf. A000009, A270913, A308680, A340987.
Sequence in context: A149796 A348410 A005191 * A275027 A147091 A149797
Adjacent sequences: A324592 A324593 A324594 * A324596 A324597 A324598
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Sep 03 2019
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STATUS
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approved
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