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A304212
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Number of partitions of n^3 into exactly n^2 parts.
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2
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1, 1, 5, 318, 112540, 139620591, 491579082022, 4303961368154069, 85434752794871493882, 3588523098005804563697043, 302194941264401427042462944147, 48844693123353655726678707534158535, 14615188708581196626576773497618986350642
(list;
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history;
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internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^(n^3-n^2)] Product_{k=1..n^2} 1/(1-x^k).
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EXAMPLE
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n | Partitions of n^3 into exactly n^2 parts
--+-------------------------------------------------
1 | 1.
2 | 5+1+1+1 = 4+2+1+1 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2.
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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(n^3-n^2, n^2):
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MATHEMATICA
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$RecursionLimit = 2000;
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1]+b[n-i, Min[i, n-i]]];
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PROG
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(PARI) {a(n) = polcoeff(prod(k=1, n^2, 1/(1-x^k+x*O(x^(n^3-n^2)))), n^3-n^2)}
(Python)
import sys
from functools import lru_cache
sys.setrecursionlimit(10**6)
@lru_cache(maxsize=None)
def b(n, i): return 1 if n == 0 or i == 1 else b(n, i-1)+b(n-i, min(i, n-i))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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