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A055507 a(n) = Sum_{k=1..n} d(k)*d(n+1-k), where d(k) is number of positive divisors of k. 23
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) 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
Convolution of A341062 and nonzero terms of A006218. - Omar E. Pol, Feb 16 2021
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.
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
a(n) = A212151(n+2) - A212151(n+1). - Ridouane Oudra, Sep 12 2020
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
CROSSREFS
Sequence in context: A317292 A276221 A265284 * A343880 A121896 A368610
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 March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)