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A055507 Sum{k = 1 to n}[d(k)*d(n+1-k)], where d(k) is number of positive divisors of k. 11
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 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

Titchmarsh, E. C. "Some problems in the analytic theory of numbers." The Quarterly Journal of Mathematics 1 (1942): 129-152.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. 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. 529-572.

FORMULA

G.f.: Sum_{i >= 1, j >= 1} x^(i+j-1)/(1-x^i)/(1-x^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(n-1)*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(n-j), j=1..n-1);

PROG

(PARI) a(n)=sum(k=1, n, numdiv(k)*numdiv(n+1-k)) \\ 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

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)