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A304134
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Number of partitions of 5n into exactly n parts.
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1
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1, 1, 5, 19, 64, 192, 532, 1367, 3319, 7657, 16928, 36043, 74287, 148702, 290071, 552767, 1031391, 1887776, 3395084, 6007963, 10474462, 18010859, 30574655, 51284587, 85064661, 139620591, 226914505, 365371100, 583164222, 923075291, 1449643115, 2259616844
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OFFSET
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0,3
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COMMENTS
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Also, the number of partitions of 4n in which every part is <=n.
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LINKS
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EXAMPLE
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n | Partitions of 5n into exactly n parts
--+------------------------------------------------
1 | 5;
2 | 9+1, 8+2, 7+3, 6+4, 5+5;
3 | 13+1+1, 12+2+1, 11+3+1, 11+2+2, 10+4+1, 10+3+2,
| 9+5+1, 9+4+2, 9+3+3, 8+6+1, 8+5+2, 8+4+3,
| 7+7+1, 7+6+2, 7+5+3, 7+4+4, 6+6+3, 6+5+4,
| 5+5+5;
====================================================================
n | Partitions of 4n in which every part is <=n.
--+-----------------------------------------------------------------
1 | 1+1+1+1;
2 | 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1;
3 | 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1,
| 3+3+2+1+1+1+1, 3+3+1+1+1+1+1+1, 3+2+2+2+2+1, 3+2+2+2+1+1+1,
| 3+2+2+1+1+1+1+1, 3+2+1+1+1+1+1+1+1, 3+1+1+1+1+1+1+1+1+1,
| 2+2+2+2+2+2, 2+2+2+2+2+1+1, 2+2+2+2+1+1+1+1, 2+2+2+1+1+1+1+1+1,
| 2+2+1+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1+1,
| 1+1+1+1+1+1+1+1+1+1+1+1;
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1) +b(n-i, min(i, n-i)))
end:
a:= n-> b(4*n, n):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]];
a[n_] := b[4n, n];
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PROG
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(PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(4*n)))), 4*n)}
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CROSSREFS
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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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