OFFSET
0,3
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.
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,
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.
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 expansion coefficient of q^4 of the second factor of the cn formula is +cos(1*v) - cos(3*v) + cos(9*v).
MATHEMATICA
Table[(-1)^(n + (# - 1)/2) # &@ Divisors[2 n + 1], {n, 0, 30}] // Flatten (* Michael De Vlieger, Aug 01 2016 *)
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Jul 27 2016
STATUS
approved