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%I #14 Aug 07 2019 15:08:14
%S 0,1,2,5,8,14,24,37,56,85,124,178,254,354,489,671,907,1217,1624,2144,
%T 2815,3675,4764,6142,7885,10062,12788,16183,20391,25590,32013,39883,
%U 49536,61326,75688,93129,114296,139856,170718,207857,252476,305938,369946,446314,537379
%N Number of triangular number parts in all partitions of n.
%C a(n) = Sum_{k=0..n} k*A263234(n,k).
%H Alois P. Heinz, <a href="/A263235/b263235.txt">Table of n, a(n) for n = 0..1000</a>
%F 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 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 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 # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p `if`(i<1, 0, b(n, i-1) +`if`(i>n, 0, (p-> p+
%p `if`(issqr(8*i+1), [0, p[1]], 0))(b(n-i, i)))))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Nov 13 2015
%t 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 Cf. A263234, A309536.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Nov 12 2015
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