A306187 - OEIS

%I #17 Sep 19 2019 10:28:55

%S 1,1,3,10,65,371,3780,33552,472971,5736082,97047819,1547576394,

%T 32992294296,626527881617,15202246707840,352290010708120,

%U 9970739854456849,262225912049078193,8309425491887714632,250946978120046026219,8898019305511325083149

%N Number of n-times partitions of n.

%C A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n. The only 0-times partition of n is the number n itself. - _Gus Wiseman_, Jan 27 2019

%H Alois P. Heinz, <a href="/A306187/b306187.txt">Table of n, a(n) for n = 0..410</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F a(n) = A323718(n,n).

%e From _Gus Wiseman_, Jan 27 2019: (Start)

%e The a(1) = 1 through a(3) = 10 partitions:

%e   (1)  ((2))     (((3)))

%e        ((11))    (((21)))

%e        ((1)(1))  (((111)))

%e                  (((2)(1)))

%e                  (((11)(1)))

%e                  (((2))((1)))

%e                  (((1)(1)(1)))

%e                  (((11))((1)))

%e                  (((1)(1))((1)))

%e                  (((1))((1))((1)))

%e (End)

%p b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,

%p       1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))

%p     end:

%p a:= n-> b(n$3):

%p seq(a(n), n=0..25);

%t ptnlevct[n_,k_]:=Switch[k,0,1,1,PartitionsP[n],_,SeriesCoefficient[Product[1/(1-ptnlevct[m,k-1]*x^m),{m,n}],{x,0,n}]];

%t Table[ptnlevct[n,n],{n,0,8}] (* _Gus Wiseman_, Jan 27 2019 *)

%Y Main diagonal of A323718.

%Y Cf. A000041, A306188.

%Y Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jan 27 2019