%I #9 Dec 10 2020 03:27:18

%S 0,1,7,25,109,315,1179,3234,10789,29517,89217,238422,708782,1838147,

%T 5158957,13489966,36783238,94122716,252156677,638254389,1674465026,

%U 4215683662,10834938301,27032106456,68911496918,170196098655,427274190051,1051046411165,2612004809769

%N Number of parts in all thrice partitions of n.

%H Alois P. Heinz, <a href="/A327627/b327627.txt">Table of n, a(n) for n = 0..2000</a>

%p b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],

%p `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+

%p (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*

%p b(n-i, min(n-i, i), k)))(b(i$2, k-1))))

%p end:

%p a:= n-> b(n$2, 3)[2]:

%p seq(a(n), n=0..30);

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/ h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];

%t a[n_] := b[n, n, 3][[2]];

%t a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)

%Y Column k=3 of A327618.

%Y Cf. A327628.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 19 2019