A304208 - OEIS

%I #24 Nov 14 2020 08:37:41

%S 1,1,3,48,1425,66055,4234086,348907094,35277846729,4236771148454,

%T 590133028697501,93613602614249377,16671698429605679621,

%U 3295006292978246618505,715884159450254458674982,169624990695197593491828744,43538384149387312404895504349

%N Number of partitions of n^3 into exactly n distinct parts.

%H Alois P. Heinz, <a href="/A304208/b304208.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = [x^(n^3-n*(n+1)/2)] Product_{k=1..n} 1/(1-x^k).

%e n | Partitions of n^3 into exactly n distinct parts

%e --+-------------------------------------------------------------

%e 1 |   1.

%e 2 |   7+1 = 6+2 = 5+3.

%e 3 |   24+ 2+1 = 23+ 3+1 = 22+ 4+1 = 22+ 3+2 = 21+ 5+1 = 21+ 4+2

%e   | = 20+ 6+1 = 20+ 5+2 = 20+ 4+3 = 19+ 7+1 = 19+ 6+2 = 19+ 5+3

%e   | = 18+ 8+1 = 18+ 7+2 = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2

%e   | = 17+ 7+3 = 17+ 6+4 = 16+10+1 = 16+ 9+2 = 16+ 8+3 = 16+ 7+4

%e   | = 16+ 6+5 = 15+11+1 = 15+10+2 = 15+ 9+3 = 15+ 8+4 = 15+ 7+5

%e   | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6

%e   | = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6 = 12+11+4

%e   | = 12+10+5 = 12+ 9+6 = 12+ 8+7 = 11+10+6 = 11+ 9+7 = 10+ 9+8.

%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+1)/2, n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, May 08 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+1)/2, n];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Nov 14 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+1)/2)))), n^3-n*(n+1)/2)}

%Y Cf. A107379, A128854, A304176.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 08 2018