

A055507


Sum{k = 1 to n}[d(k)*d(n+1k)], where d(k) is number of positive divisors of k.


8



1, 4, 8, 14, 20, 28, 37, 44, 58, 64, 80, 86, 108, 108, 136, 134, 169, 160, 198, 192, 236, 216, 276, 246, 310, 288, 348, 310, 400, 344, 433, 396, 474, 408, 544, 450, 564, 512, 614, 522, 688, 560, 716, 638, 756, 636, 860, 676, 859, 772, 926, 758, 1016, 804, 1032
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OFFSET

1,2


COMMENTS

a(n) = number of ordered ways to express n+1 as a*b+c*d with 1 <= a,b,c,d <= n.  David W. Wilson, Jun 16 2003
tau(n) (A000005) convolved with itself, treating this result as a sequence whose offset is 2.  Graeme McRae, Jun 06 2006


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115130. See D_{0,0}.
Yoichi Motohashi, The binary additive divisor problem, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 27 no. 5 (1994), p. 529572.


FORMULA

G.f.: Sum_{i >= 1, j >= 1} x^(i+j1)/(1x^i)/(1x^j).  Vladeta Jovovic, Nov 11 2001
Working with an offset of 2, it appears that the o.g.f is equal to the Lambert series sum {n >= 2} A072031(n1)*x^n/(1  x^n).  Peter Bala, Dec 09 2014


EXAMPLE

a[4] = d(1)*d(4) + d(2)*d(3) + d(3)*d(2) + d(4)*d(1) = 1*3 +2*2 +2*2 +3*1 = 14
3 = 1*1+2*1 in 4 ways, so a(2)=4; 4 = 1*1+1*3 (4 ways) = 2*1+2*1 (4 ways), so a(3)=8; 5 = 4*1+1*1 (4 ways) = 2*2+1*1 (2 ways) + 3*1+2*1 (8 ways), so a(4) = 14.  N. J. A. Sloane, Jul 07 2012


MAPLE

with(numtheory); D00:=n>add(tau(j)*tau(nj), j=1..n1);


PROG

(PARI) a(n)=sum(k=1, n, numdiv(k)*numdiv(n+1k)) \\ Charles R Greathouse IV, Oct 17 2012


CROSSREFS

Cf. A000385, A072031.
Sequence in context: A317292 A276221 A265284 * A121896 A173290 A312686
Adjacent sequences: A055504 A055505 A055506 * A055508 A055509 A055510


KEYWORD

nonn


AUTHOR

Leroy Quet, Jun 29 2000


EXTENSIONS

More terms from James A. Sellers, Jul 04 2000
Definition clarified by N. J. A. Sloane, Jul 07 2012


STATUS

approved



