|
|
A007437
|
|
Inverse Moebius transform of triangular numbers.
(Formerly M3309)
|
|
32
|
|
|
1, 4, 7, 14, 16, 31, 29, 50, 52, 74, 67, 119, 92, 137, 142, 186, 154, 247, 191, 294, 266, 323, 277, 455, 341, 446, 430, 553, 436, 686, 497, 714, 634, 752, 674, 1001, 704, 935, 878, 1150, 862, 1298, 947, 1323, 1222, 1361, 1129, 1767, 1254, 1674, 1486, 1834
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
|
|
FORMULA
|
Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^3.
G.f.: Sum_{n>=1} (n*(n+1)/2) * x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^((k+1)/2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
|
|
MAPLE
|
with(numtheory):
a:= proc(n) option remember;
add(d*(d+1)/2, d=divisors(n))
end:
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) a(n)=if(n<1, 1, sumdiv(n, d, (d^2+d))/2); /* Joerg Arndt, Aug 14 2012 */
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|