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A034715 Dirichlet convolution of triangular numbers with themselves.

%I A034715 #16 Sep 03 2025 00:51:49
%S A034715 1,6,12,29,30,78,56,132,126,200,132,402,182,378,420,588,306,864,380,
%T A034715 1050,798,902,552,1920,875,1248,1296,2002,870,2940,992,2592,1914,2108,
%U A034715 2100,4635,1406,2622,2652,5080,1722,5628,1892,4818,4860,3818,2256,8856
%N A034715 Dirichlet convolution of triangular numbers with themselves.
%H A034715 Bruno Berselli, <a href="/A034715/b034715.txt">Table of n, a(n) for n = 1..1000</a>
%F A034715 G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^k/(1 - x^k)^3. - _Ilya Gutkovskiy_, Oct 24 2018
%F A034715 From _Vaclav Kotesovec_, Feb 05 2019: (Start)
%F A034715 Dirichlet g.f.: ((zeta(s-1) + zeta(s-2))/2)^2.
%F A034715 Sum_{k=1..n} a(k) ~ n^3*(log(n)/12 + (6*gamma - 1 + Pi^2)/36), where gamma is the Euler-Mascheroni constant A001620. (End)
%t A034715 Table[n/4*Sum[(n+d)*(d+1)/d, {d, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, Feb 05 2019 *)
%o A034715 (Magma) A000217:=func<i | i*(i+1)/2>; [&+[A000217(d)*A000217(n div d): d in Divisors(n)]: n in [1..50]]; // _Bruno Berselli_, Feb 11 2014
%Y A034715 Cf. A000217.
%K A034715 nonn,changed
%O A034715 1,2
%A A034715 _Erich Friedman_