OFFSET
0,3
COMMENTS
The length of row n is A099774(n+1).
This gives the odd numbered rows of the irregular triangle A027750.
The row sums are given in A008438.
The entries of row n appear, for instance, as arguments of sin in the Fourier expansion of Jacobi's elliptic function sn in the second factor Sum_{n>=0} (q^n/(1-q^(2*n+1)))*sin((2*n+1)*v) as coefficients of q^n. See e.g., the formula in Abramowitz-Stegun, p. 575, 16.23.1 (or 16.23.2 for cn but with signs). See also A274659.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972,
FORMULA
T(n, k) = k-th divisor of 2*n+1 in increasing order.
EXAMPLE
The irregular triangle T(n, k) begins:
n, 2n+1\k 1 2 3 4 ...
0, 1: 1
1, 3: 1 3
2, 5: 1 5
3, 7: 1 7
4, 9: 1 3 9
5, 11: 1 11
6, 13: 1 13
7, 15: 1 3 5 15
8, 17: 1 17
9, 19: 1 19
10, 21: 1 3 7 21
11, 23: 1 23
12, 25: 1 5 25
13, 27: 1 3 9 27
14, 29: 1 29
15, 31: 1 31
16, 33: 1 3 11 33
17, 35: 1 5 7 35
18, 37: 1 37
19, 39: 1 3 13 39
20, 41: 1 41
...
The above mentioned second factor in the sn formula has as q^4 coefficient: sin(1*v) + sin(3*v) + sin(9*v).
MATHEMATICA
Table[Divisors[2 n + 1], {n, 0, 22}] // Flatten (* Michael De Vlieger, Jul 18 2016 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Wolfdieter Lang, Jul 18 2016
STATUS
approved