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A246721 Number of partitions of n into parts of the n-th list of distinct parts in the order given by A246688.

%I A246721 #12 Dec 07 2020 10:07:02
%S A246721 1,1,1,2,0,2,0,2,2,0,14,3,4,0,20,3,2,1,0,26,24,4,4,2,1,35,31,4,24,2,6,
%T A246721 1,0,378,54,42,42,5,31,0,2,0,0,631,78,61,56,5,45,34,3,3,2,2,0,1045,
%U A246721 992,110,85,75,73,6,55,0,7,42,8,0,2,0,1772,1581,156
%N A246721 Number of partitions of n into parts of the n-th list of distinct parts in the order given by A246688.
%C A246721 The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .
%H A246721 Alois P. Heinz, <a href="/A246721/b246721.txt">Table of n, a(n) for n = 0..5000</a>
%F A246721 a(n) = A246720(n,n).
%e A246721 a(7) = 2 because there are 2 partitions of 7 into parts 1, 4: [1,1,1,1,1,1,1], [1,1,1,4].
%p A246721 b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
%p A246721       [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
%p A246721     end:
%p A246721 f:= proc() local i, l; i, l:=0, [];
%p A246721       proc(n) while n>=nops(l)
%p A246721         do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
%p A246721       end
%p A246721     end():
%p A246721 g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
%p A246721       add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
%p A246721     end:
%p A246721 a:= n-> g(n, f(n)):
%p A246721 seq(a(n), n=0..80);
%t A246721 b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
%t A246721 f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
%t A246721 g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
%t A246721 a[n_] := g[n, f[n]];
%t A246721 a /@ Range[0, 80] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A246721 Main diagonal of A246720.
%Y A246721 Cf. A246688, A246691 (the same for compositions).
%K A246721 nonn,look
%O A246721 0,4
%A A246721 _Alois P. Heinz_, Sep 02 2014