A087115 - OEIS

%I #27 Jan 11 2025 03:11:00

%S 0,1,18,137,650,2350,6860,17609,39870,83976,162382,301070,522886,

%T 885284,1424468,2254537,3419448,5143987,7448874,10750712,15015872,

%U 20948610,28373444,38539022,50863150,67454492,87209316,113326308,143748766,183759900,229271536

%N Convolution of sum of cubes of divisors with itself.

%C Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

%D Jean-Pierre Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chapter VII, Section 4., p. 93.

%H Seiichi Manyama, <a href="/A087115/b087115.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (Sum_{k>0} k^3 * x^k / (1 - x^k))^2.

%F a(n) = (sigma_7(n) - sigma_3(n)) / 120.

%F G.f.: ((Q(x) - 1) / 240)^2 where Q() is a Ramanujan Eisenstein series.

%F Dirichlet g.f.: zeta(s) * (zeta(s-7) - zeta(s-3)) / 120. - _Amiram Eldar_, Jan 11 2025

%e G.f. = x^2 + 18*x^3 + 137*x^4 + 650*x^5 + 2350*x^6 + 6860*x^7 + 17609*x^8 + ...

%p with(numtheory); f:=n->add( sigma[3](k)*sigma[3](n-k),k=1..n-1);

%t a[ n_] := If[ n < 1, 0, (DivisorSigma[ 7, n] - DivisorSigma[ 3, n]) / 120]; (* _Michael Somos_, Oct 08 2017 *)

%o (PARI) {a(n) = if( n<1, 0, (sigma(n, 7) - sigma(n, 3)) / 120)};

%o (PARI) {a(n) = if( n<1, 0, sum(m=1, n-1, sigma(m, 3) * sigma(n-m, 3)))};

%Y Cf. A004009.

%Y Cf. A001158 (sigma_3), A013955 (sigma_7). [_Ridouane Oudra_, Apr 22 2020]

%K nonn

%O 1,3

%A _Michael Somos_, Aug 13 2003