A191831 - OEIS

%I #37 Dec 13 2021 17:25:39

%S 0,1,5,12,24,39,60,87,113,158,189,249,286,372,402,516,545,696,709,886,

%T 912,1125,1110,1401,1348,1674,1654,1992,1906,2390,2226,2735,2648,3141,

%U 2926,3705,3346,4069,3898,4604,4223,5282,4707,5757,5426,6326,5754,7269,6324,7669,7230,8468,7556,9456,8240,10018,9320,10748,9621,12246

%N a(n) = Sum_{i+j=n, i,j >= 1} tau(i)*sigma(j), where tau() = A000005(), sigma() = A000203().

%C This is Andrews's D_{0,1}(n).

%C From _Omar E. Pol_, Dec 08 2021: (Start)

%C Zero together with the convolution of A000005 and A000203.

%C Zero together with the convolution of A341062 and A024916.

%C Zero together with the convolution of the nonzero terms of A006218 and A340793.

%C a(n) is also the volume of a symmetric polycube which belongs to the family of symmetric polycubes that represent the convolution of A000203 with any other integer sequence, n >= 1. (End)

%H George E. Andrews, <a href="http://dx.doi.org/10.1007/BF01608779">Stacked lattice boxes</a>, Ann. Comb. 3 (1999), 115-130.

%F G.f.: (Sum_{k>=1} x^k/(1 - x^k))*(Sum_{k>=1} k*x^k/(1 - x^k)). - _Ilya Gutkovskiy_, Jan 01 2017

%p with(numtheory); D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);

%p [seq(D01(n),n=1..60)];

%t Table[Sum[DivisorSigma[0, j] DivisorSigma[1, n - j], {j, n - 1}], {n, 60}] (* _Michael De Vlieger_, Jan 01 2017 *)

%o (PARI) a(n)=sum(i=1,n-1,numdiv(i)*sigma(n-i)) \\ _Charles R Greathouse IV_, Feb 19 2013

%Y Cf. A000005, A000203 A006218, A024916, A086718, A237593, A340793, A341062.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, Jun 17 2011