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A274660
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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).
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2
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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
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OFFSET
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0,3
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COMMENTS
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The length of row n is A099774(n+1).
The unsigned irregular triangle is given in A274658.
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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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MATHEMATICA
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Table[(-1)^(n + (# - 1)/2) # &@ Divisors[2 n + 1], {n, 0, 30}] // Flatten (* Michael De Vlieger, Aug 01 2016 *)
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CROSSREFS
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KEYWORD
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sign,easy,tabf
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AUTHOR
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STATUS
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approved
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