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

Author

Wolfdieter Lang, Jul 18 2016

Keywords

• nonn
• tabf
• easy

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.

Examples

			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).
		

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972,

Crossrefs

Cf. A008438 (row sums), A027750, A099774 (row lengths), A274659.

Programs

Mathematica

Table[Divisors[2 n + 1], {n, 0, 22}] // Flatten (* Michael De Vlieger, Jul 18 2016 *)

PARI

row(n) = divisors(2*n+1); \\ Amiram Eldar, May 02 2025

Formula

T(n, k) = k-th divisor of 2*n+1 in increasing order.