A332719 - OEIS

%I #14 Jan 06 2021 02:12:39

%S 1,3,8,19,44,93,187,357,657,1166,2015,3393,5594,9044,14378,22501,

%T 34734,52931,79735,118823,175337,256347,371606,534377,762721,1080979,

%U 1521925,2129330,2961580,4096006,5634855,7712558,10505457,14243772,19227383,25845241,34600673

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

%C The canonical ordering of partitions is described in A080577.

%H Alois P. Heinz, <a href="/A332719/b332719.txt">Table of n, a(n) for n = 0..4000</a>

%H Wikipedia, <a href="https://www.wikipedia.org/wiki/integer_partition">Integer Partition</a>

%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*sqrt(n)). - _Vaclav Kotesovec_, Feb 28 2020

%e a(2) = 8, because 222 has position 8 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .

%p b:= proc(n, i) option remember;

%p      `if`(n=0, 1, b(n-i, i)+g(n, i-1))

%p     end:

%p g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

%p      `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))

%p     end:

%p a:= n-> g(3*n$2)-b(3*n, n)+1:

%p seq(a(n), n=0..37);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i] + g[n, i - 1]];

%t 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 a[n_] := g[3n, 3n] - b[3n, n] + 1;

%t a /@ Range[0, 37] (* _Jean-François Alcover_, Jan 06 2021, after _Alois P. Heinz_ *)

%Y Cf. A000041, A080577, A322761, A330661, A332706, A332720.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Feb 20 2020