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A306187 Number of n-times partitions of n.

%I A306187 #17 Sep 19 2019 10:28:55
%S A306187 1,1,3,10,65,371,3780,33552,472971,5736082,97047819,1547576394,
%T A306187 32992294296,626527881617,15202246707840,352290010708120,
%U A306187 9970739854456849,262225912049078193,8309425491887714632,250946978120046026219,8898019305511325083149
%N A306187 Number of n-times partitions of n.
%C A306187 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 A306187 Alois P. Heinz, <a href="/A306187/b306187.txt">Table of n, a(n) for n = 0..410</a>
%H A306187 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A306187 a(n) = A323718(n,n).
%e A306187 From _Gus Wiseman_, Jan 27 2019: (Start)
%e A306187 The a(1) = 1 through a(3) = 10 partitions:
%e A306187   (1)  ((2))     (((3)))
%e A306187        ((11))    (((21)))
%e A306187        ((1)(1))  (((111)))
%e A306187                  (((2)(1)))
%e A306187                  (((11)(1)))
%e A306187                  (((2))((1)))
%e A306187                  (((1)(1)(1)))
%e A306187                  (((11))((1)))
%e A306187                  (((1)(1))((1)))
%e A306187                  (((1))((1))((1)))
%e A306187 (End)
%p A306187 b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
%p A306187       1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p A306187     end:
%p A306187 a:= n-> b(n$3):
%p A306187 seq(a(n), n=0..25);
%t A306187 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 A306187 Table[ptnlevct[n,n],{n,0,8}] (* _Gus Wiseman_, Jan 27 2019 *)
%Y A306187 Main diagonal of A323718.
%Y A306187 Cf. A000041, A306188.
%Y A306187 Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.
%K A306187 nonn
%O A306187 0,3
%A A306187 _Alois P. Heinz_, Jan 27 2019