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A332720 Index position of {3}^n within the list of partitions of 3n in canonical ordering.

%I A332720 #16 Jan 06 2021 02:12:31
%S A332720 1,1,5,19,59,150,349,745,1515,2936,5514,10036,17851,31039,53006,88943,
%T A332720 147057,239701,385885,613855,966137,1505137,2323124,3553914,5392315,
%U A332720 8117758,12131618,18003740,26543030,38886999,56633453,82009410,118113488,169229009,241264461
%N A332720 Index position of {3}^n within the list of partitions of 3n in canonical ordering.
%C A332720 The canonical ordering of partitions is described in A080577.
%H A332720 Alois P. Heinz, <a href="/A332720/b332720.txt">Table of n, a(n) for n = 0..4000</a>
%H A332720 Wikipedia, <a href="https://www.wikipedia.org/wiki/integer_partition">Integer Partition</a>
%F A332720 a(n) ~ exp(Pi*sqrt(2*n)) / (4*3^(3/2)*n). - _Vaclav Kotesovec_, Feb 28 2020
%e A332720 a(2) = 5, because 33 has position 5 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .
%p A332720 b:= proc(n) option remember;
%p A332720      `if`(n=0, 1, b(n-1)+g(3*n, 2))
%p A332720     end:
%p A332720 g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A332720      `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
%p A332720     end:
%p A332720 a:= n-> g(3*n$2)-b(n)+1:
%p A332720 seq(a(n), n=0..35);
%t A332720 b[n_] := b[n] = If[n == 0, 1, b[n - 1] + g[3n, 2]];
%t A332720 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 A332720 a[n_] := g[3n, 3n] - b[n] + 1;
%t A332720 a /@ Range[0, 35] (* _Jean-François Alcover_, Jan 06 2021, after _Alois P. Heinz_ *)
%Y A332720 Cf. A000041, A080577, A322761, A330661, A332706, A332719.
%K A332720 nonn
%O A332720 0,3
%A A332720 _Alois P. Heinz_, Feb 20 2020