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%I #48 Sep 09 2021 16:30:17
%S 1,1,4,61,1906,91606,6023602,505853354,51900711796,6306147384659,
%T 886745696653253,141778041323736643,25417656781153090889,
%U 5052180112449982704619,1103058286595668300801794,262487324530101028337614478,67628783852463631751658038290
%N Number of partitions of n^3 into exactly n parts.
%H Chai Wah Wu, <a href="/A304176/b304176.txt">Table of n, a(n) for n = 0..136</a> (terms 0..96 from Alois P. Heinz)
%F a(n) = [x^(n^3-n)] Product_{k=1..n} 1/(1-x^k).
%e n | Partitions of n^3 into exactly n parts
%e --+------------------------------------------------------------
%e 1 | 1.
%e 2 | 7+1 = 6+2 = 5+3 = 4+4.
%e 3 | 25+ 1+1 = 24+ 2+1 = 23+ 3+1 = 23+ 2+2 = 22+ 4+1 = 22+ 3+2
%e | = 21+ 5+1 = 21+ 4+2 = 21+ 3+3 = 20+ 6+1 = 20+ 5+2 = 20+ 4+3
%e | = 19+ 7+1 = 19+ 6+2 = 19+ 5+3 = 19+ 4+4 = 18+ 8+1 = 18+ 7+2
%e | = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2 = 17+ 7+3 = 17+ 6+4
%e | = 17+ 5+5 = 16+10+1 = 16+ 9+2 = 16+ 8+3 = 16+ 7+4 = 16+ 6+5
%e | = 15+11+1 = 15+10+2 = 15+ 9+3 = 15+ 8+4 = 15+ 7+5 = 15+ 6+6
%e | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6
%e | = 13+13+1 = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6
%e | = 13+ 7+7 = 12+12+3 = 12+11+4 = 12+10+5 = 12+ 9+6 = 12+ 8+7
%e | = 11+11+5 = 11+10+6 = 11+ 9+7 = 11+ 8+8 = 10+10+7 = 10+ 9+8
%e | = 9+ 9+9.
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p b(n, i-1)+b(n-i, min(i, n-i)))
%p end:
%p a:= n-> b(n^3-n, n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 07 2018
%t $RecursionLimit = 2000;
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + b[n - i, Min[i, n - i]]];
%t a[n_] := b[n^3 - n, n];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%o (PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^3-n)))), n^3-n)}
%o (Python)
%o import sys
%o from functools import lru_cache
%o sys.setrecursionlimit(10**6)
%o @lru_cache(maxsize=None)
%o def b(n,i): return 1 if n == 0 or i == 1 else b(n,i-1)+b(n-i,min(i,n-i))
%o def A304176(n): return b(n**3-n,n) # _Chai Wah Wu_, Sep 09 2021, after _Alois P. Heinz_
%Y Cf. A206240, A238608, A304208, A304212.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 07 2018
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