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A368638 a(n) is the number of triangular partitions whose Young diagram fits inside a square of side n.

%I A368638 #7 Jan 06 2024 09:20:39
%S A368638 1,2,5,12,25,48,83,136,211,314,449,626,849,1130,1475,1892,2389,2982,
%T A368638 3677,4492,5435,6518,7751,9156,10741,12526,14523,16750,19219,21958,
%U A368638 24975,28300,31949,35942,40295,45032,50165,55730,61745,68234,75213,82722,90773,99408
%N A368638 a(n) is the number of triangular partitions whose Young diagram fits inside a square of side n.
%C A368638 Equivalently, a(n) is the number of triangular subpartitions of the staircase partition (n, n-1, ..., 1).
%H A368638 Sergi Elizalde and Alejandro B. Galván, <a href="https://arxiv.org/abs/2312.16353">Triangular partitions: enumeration, structure, and generation</a>, arXiv:2312.16353 [math.CO], (2023).
%F A368638 a(n) = 1 + Sum_{i=1..n} binomial(n-i+2,2)*phi(i).
%o A368638 (MATLAB)
%o A368638 % subpart(n) := a(n-1).
%o A368638 nmax = 44;
%o A368638 for n = 1 : nmax
%o A368638     subpart(n) = 1;
%o A368638     for i = 1 : n
%o A368638         subpart(n) = subpart(n) + (n - i + 1)*(n - i)*eulerPhi(i)/2;
%o A368638     end
%o A368638 end
%Y A368638 The number of triangular partitions of size n is in A352882.
%K A368638 nonn
%O A368638 0,2
%A A368638 _Alejandro B. Galván_, Jan 01 2024