cp's OEIS Frontend

A332722 Index position of [2n-1, 2n-3, ..., 3, 1] within the list of partitions of n^2 in canonical ordering.

%I A332722 #17 May 10 2020 03:24:40
%S A332722 1,1,2,9,74,711,7312,77793,848557,9426039,106218592,1210785512,
%T A332722 13933358426,161624712815,1887635428421,22176331059637,
%U A332722 261881397819259,3106736469937751,37006306302036790,442425926101676831,5306994321265281854,63851605555921588684,770371217568310624912
%N A332722 Index position of [2n-1, 2n-3, ..., 3, 1] within the list of partitions of n^2 in canonical ordering.
%C A332722 The canonical ordering of partitions is described in A080577.
%H A332722 Alois P. Heinz, <a href="/A332722/b332722.txt">Table of n, a(n) for n = 0..130</a>
%H A332722 Wikipedia, <a href="https://www.wikipedia.org/wiki/integer_partition">Integer Partition</a>
%e A332722 a(3) = 9, because 531 has position 9 within the list of partitions of 3*3 in canonical ordering: 9, 81, 72, 711, 63, 621, 6111, 54, 531, ... .
%p A332722 b:= proc(n, i) option remember;
%p A332722      `if`(n=0, 1, b(n-i, i-2)+g(n, i-1))
%p A332722     end:
%p A332722 g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A332722      `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
%p A332722     end:
%p A332722 a:= n-> g(n^2$2)-b(n^2, 2*n-1)+1:
%p A332722 seq(a(n), n=0..23);
%t A332722 b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i - 2] + g[n, i - 1]];
%t A332722 g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
%t A332722 a[n_] := g[n^2, n^2] - b[n^2, 2n - 1] + 1;
%t A332722 a /@ Range[0, 23] (* _Jean-François Alcover_, May 10 2020, after Maple *)
%Y A332722 Cf. A000041, A000290, A005408, A080577, A238639, A238640, A322761.
%K A332722 nonn
%O A332722 0,3
%A A332722 _Alois P. Heinz_, Feb 20 2020