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%I M5193 N2257 #44 Aug 08 2022 08:38:30
%S 0,1,27,184,875,2700,7546,17600,35721,72750,126445,223776,353717,
%T 595448,843750,1349120,1827636,2808837,3600975,5306000,6667920,
%U 9599172,11509982,16416000,19015625,26605670,30902310,41686848,46948825,64233000,70306760,94089216
%N a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
%H John Cerkan, <a href="/A000499/b000499.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F a(n) = Sum_{k=1..n-1} k^3*sigma(k)*sigma(n-k). - _Michel Marcus_, Feb 02 2014
%F a(n) = (n^3/24 - n^4/8)*sigma_1(n) + (n^3/12)*sigma_3(n). - _Ridouane Oudra_, Sep 15 2020
%F Sum_{k=1..n} a(k) ~ Pi^4 * n^7 / 7560. - _Vaclav Kotesovec_, Aug 08 2022
%e G.f. = x^2 + 27*x^3 + 184*x^4 + 875*x^5 + 2700*x^6 + 7546*x^7 + 17600*x^8 + ...
%p S:=(n,e)->add(k^e*sigma(k)*sigma(n-k),k=1..n-1); f:=e->[seq(S(n,e),n=1..30)]; f(3);
%t a[n_] := Sum[k^3*DivisorSigma[1, k]*DivisorSigma[1, n - k], {k, 1, n - 1}]; Array[a, 32] (* _Jean-François Alcover_, Feb 09 2016 *)
%o (PARI) a(n) = sum(k=1, n-1, k^3*sigma(k)*sigma(n-k)); \\ _Michel Marcus_, Feb 02 2014
%Y Cf. A000385, A000441, A000477, A259692, A259693, A259694, A259695, A259696.
%Y Cf. A000203 (sigma_1), A001158 (sigma_3).
%K nonn
%O 1,3
%A _N. J. A. Sloane_
%E More terms and 0 prepended by _Michel Marcus_, Feb 02 2014
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