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A263235 Number of triangular number parts in all partitions of n.

%I A263235 #17 Jun 12 2025 10:23:33
%S A263235 0,1,2,5,8,14,24,37,56,85,124,178,254,354,489,671,907,1217,1624,2144,
%T A263235 2815,3675,4764,6142,7885,10062,12788,16183,20391,25590,32013,39883,
%U A263235 49536,61326,75688,93129,114296,139856,170718,207857,252476,305938,369946,446314,537379
%N A263235 Number of triangular number parts in all partitions of n.
%C A263235 a(n) = Sum_{k=0..n} k*A263234(n,k).
%H A263235 Vaclav Kotesovec, <a href="/A263235/b263235.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A263235 G.f.: Sum_{i>0} x^h(i)/(1-x^h(i)) / Product_{i>0} 1-x^i, where h(i) = i*(i+1)/2.
%e A263235 a(4) = 8 because the partitions of 4 are  [4], [3',1'], [2,2], [2,1',1'], and [1',1',1',1'], where the triangular number parts are marked.
%p A263235 h:= proc (i) options operator, arrow: (1/2)*i*(i+1) end proc: g := (sum(x^h(i)/(1-x^h(i)), i = 1..100))/(product(1-x^i, i = 1..100)): hser:= series(g, x = 0, 55): seq(coeff(hser, x, n), n = 0..50);
%p A263235 # second Maple program:
%p A263235 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A263235       `if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, (p-> p+
%p A263235       `if`(issqr(8*i+1), [0, p[1]], 0))(b(n-i, i)))))
%p A263235     end:
%p A263235 a:= n-> b(n$2)[2]:
%p A263235 seq(a(n), n=0..60);  # _Alois P. Heinz_, Nov 13 2015
%t A263235 b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, p + If[IntegerQ@Sqrt[8*i+1], {0, p[[1]]}, 0]][b[n-i, i]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 08 2017, after _Alois P. Heinz_ *)
%Y A263235 Cf. A263234, A309536.
%K A263235 nonn
%O A263235 0,3
%A A263235 _Emeric Deutsch_, Nov 12 2015