A274660 Irregular triangle read by rows in which row n lists the divisors d of 2*n+1 (A274658), given the sign (-1)^(n + (d-1)/2).

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, -1, 47, 1, -7, 49, -1, 3, -17, 51, 1, 53, -1, -5, 11, 55, 1, -3, -19, 57, -1, 59, 1, 61

Offset: 0

Author

Wolfdieter Lang, Jul 27 2016

Keywords

• sign
• easy
• tabf

Comments

The length of row n is A099774(n+1).

The unsigned irregular triangle is given in A274658.

The sum of row n gives A228443(n).

The entries of row n appear in the Fourier expansion of Jacobi's elliptic function cn in the rewritten second factor Sum_{n>=0} (q^n/(1+q^(2*n+1))) * cos((2*n+1)*v) as Sum_{n>=0} q^n*Sum_{k=1..A099774(n+1)} sign(a(n,k))*cos(abs(a(n,k))*v). See e.g., the formula in Abramowitz-Stegun, p. 575, 16.23.2.

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 expansion coefficient of q^4 of the second factor of the cn formula is +cos(1*v) - cos(3*v) + cos(9*v).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..12574 (0 <= n <= 2500)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972,

Crossrefs

Cf. A099774, A228443, A274658.

Programs

Mathematica

Table[(-1)^(n + (# - 1)/2) # &@ Divisors[2 n + 1], {n, 0, 30}] // Flatten (* Michael De Vlieger, Aug 01 2016 *)

Formula

T(n, k) = (-1)^(n + (d(k)-1)/2)*d(k) with d(k) the k-th divisor of 2*n+1 in increasing order.