OFFSET
1,2
COMMENTS
a(n) is the 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), 115-130. See D_{0,0}.
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.
Yoichi Motohashi, The binary additive divisor problem, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 27 no. 5 (1994), p. 529-572.
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.
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); A055507:=n->add(tau(j)*tau(n+1-j), j=1..n);
MATHEMATICA
Table[Sum[DivisorSigma[0, k]*DivisorSigma[0, n + 1 - k], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 08 2022 *)
PROG
(PARI) a(n)=sum(k=1, n, numdiv(k)*numdiv(n+1-k)) \\ Charles R Greathouse IV, Oct 17 2012
(Python)
from sympy import divisor_count
def A055507(n): return (sum(divisor_count(i+1)*divisor_count(n-i) for i in range(n>>1))<<1)+(divisor_count(n+1>>1)**2 if n&1 else 0) # Chai Wah Wu, Jul 26 2024
CROSSREFS
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