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A304212 Number of partitions of n^3 into exactly n^2 parts.

%I A304212 #21 Sep 09 2021 15:45:04
%S A304212 1,1,5,318,112540,139620591,491579082022,4303961368154069,
%T A304212 85434752794871493882,3588523098005804563697043,
%U A304212 302194941264401427042462944147,48844693123353655726678707534158535,14615188708581196626576773497618986350642
%N A304212 Number of partitions of n^3 into exactly n^2 parts.
%H A304212 Chai Wah Wu, <a href="/A304212/b304212.txt">Table of n, a(n) for n = 0..50</a> (terms n = 0..30 from Seiichi Manyama)
%F A304212 a(n) = [x^(n^3-n^2)] Product_{k=1..n^2} 1/(1-x^k).
%e A304212 n | Partitions of n^3 into exactly n^2 parts
%e A304212 --+-------------------------------------------------
%e A304212 1 | 1.
%e A304212 2 | 5+1+1+1 = 4+2+1+1 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2.
%p A304212 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A304212       b(n, i-1)+b(n-i, min(i, n-i)))
%p A304212     end:
%p A304212 a:= n-> b(n^3-n^2, n^2):
%p A304212 seq(a(n), n=0..15);  # _Alois P. Heinz_, May 08 2018
%t A304212 $RecursionLimit = 2000;
%t A304212 b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1]+b[n-i, Min[i, n-i]]];
%t A304212 a[n_] := b[n^3 - n^2, n^2]; a /@ Range[0, 15] (* _Jean-François Alcover_, Nov 15 2020, after _Alois P. Heinz_ *)
%o A304212 (PARI) {a(n) = polcoeff(prod(k=1, n^2, 1/(1-x^k+x*O(x^(n^3-n^2)))), n^3-n^2)}
%o A304212 (Python)
%o A304212 import sys
%o A304212 from functools import lru_cache
%o A304212 sys.setrecursionlimit(10**6)
%o A304212 @lru_cache(maxsize=None)
%o A304212 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 A304212 def A304212(n): return b(n**3-n**2,n**2) # _Chai Wah Wu_, Sep 09 2021, after _Alois P. Heinz_
%Y A304212 Cf. A128854, A304176.
%K A304212 nonn
%O A304212 0,3
%A A304212 _Seiichi Manyama_, May 08 2018