A274658 Irregular triangle which lists in row n the divisors of 2*n+1.

1, 1, 3, 1, 5, 1, 7, 1, 3, 9, 1, 11, 1, 13, 1, 3, 5, 15, 1, 17, 1, 19, 1, 3, 7, 21, 1, 23, 1, 5, 25, 1, 3, 9, 27, 1, 29, 1, 31, 1, 3, 11, 33, 1, 5, 7, 35, 1, 37, 1, 3, 13, 39, 1, 41, 1, 43, 1, 3, 5, 9, 15, 45

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

Table of n, a(n) for n=0..62.

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

Cf. A027750, A099774, A274659.

Sequence in context: A309391 A147661 A155457 * A274660 A327531 A327514

Adjacent sequences: A274655 A274656 A274657 * A274659 A274660 A274661

Keyword

nonn,tabf,easy

Author

Wolfdieter Lang, Jul 18 2016

Status

approved