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A219347 Number of partitions of n into distinct parts with smallest possible largest part.

%I A219347 #22 Dec 10 2020 03:27:34
%S A219347 1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,3,2,2,1,1,1,4,3,2,2,1,1,1,5,4,3,2,2,
%T A219347 1,1,1,6,5,4,3,2,2,1,1,1,8,6,5,4,3,2,2,1,1,1,10,8,6,5,4,3,2,2,1,1,1,
%U A219347 12,10,8,6,5,4,3,2,2,1,1,1,15,12,10,8,6,5
%N A219347 Number of partitions of n into distinct parts with smallest possible largest part.
%C A219347 Size of the smallest possible largest part is floor(sqrt(2*n)+1/2) = A002024(n).  Records occur at 0, 7, and A000124(k) for k>=5.
%H A219347 Alois P. Heinz, <a href="/A219347/b219347.txt">Table of n, a(n) for n = 0..10000</a>
%e A219347 a(0) = 1: [].
%e A219347 a(7) = 2: [4,2,1], [4,3].
%e A219347 a(16) = 3: [6,4,3,2,1], [6,5,3,2], [6,5,4,1].
%e A219347 a(22) = 4: [7,5,4,3,2,1], [7,6,4,3,2], [7,6,5,3,1], [7,6,5,4].
%p A219347 g:= proc(n, i) option remember; local s; s:=i*(i+1)/2;
%p A219347       `if`(n=s, 1, `if`(n>s, 0, g(n, i-1)+ `if`(i>n, 0, g(n-i, i-1))))
%p A219347     end:
%p A219347 a:= n-> g(n, floor(sqrt(2*n)+1/2)):
%p A219347 seq (a(n), n=0..120);
%t A219347 g[n_, i_] := g[n, i] = Module[{s = i(i+1)/2}, If[n == s, 1, If[n > s, 0, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]]];
%t A219347 a[n_] := g[n, Floor[Sqrt[2n] + 1/2]];
%t A219347 a /@ Range[0, 120] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y A219347 Cf. A000009 (records), A219339.
%K A219347 nonn,look
%O A219347 0,8
%A A219347 _Alois P. Heinz_, Nov 18 2012