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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).

%I A274660 #18 Aug 13 2023 17:32:53
%S A274660 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,
%T A274660 -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,
%U A274660 -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
%N 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).
%C A274660 The length of row n is A099774(n+1).
%C A274660 The unsigned irregular triangle is given in A274658.
%C A274660 The sum of row n gives A228443(n).
%C A274660 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.
%H A274660 Michael De Vlieger, <a href="/A274660/b274660.txt">Table of n, a(n) for n = 0..12574</a> (0 <= n <= 2500)
%H A274660 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972,
%F A274660 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.
%e A274660 The irregular triangle T(n, k) begins:
%e A274660 n, 2n+1\k  1  2   3   4 ...
%e A274660 0,   1:    1
%e A274660 1,   3:   -1  3
%e A274660 2,   5:    1  5
%e A274660 3,   7:   -1  7
%e A274660 4,   9:    1 -3   9
%e A274660 5,  11:   -1 11
%e A274660 6,  13:    1 13
%e A274660 7,  15:   -1  3  -5  15
%e A274660 8,  17:    1 17
%e A274660 9,  19:   -1 19
%e A274660 10, 21:    1 -3  -7  21
%e A274660 11, 23:   -1 23
%e A274660 12, 25:    1  5  25
%e A274660 13, 27:   -1  3  -9  27
%e A274660 14, 29:    1 29
%e A274660 15, 31:   -1 31
%e A274660 16, 33:    1 -3 -11  33
%e A274660 17, 35:   -1 -5   7  35
%e A274660 18, 37:    1 37
%e A274660 19, 39:   -1  3 -13  39
%e A274660 20, 41:    1 41
%e A274660 ...
%e A274660 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).
%t A274660 Table[(-1)^(n + (# - 1)/2) # &@ Divisors[2 n + 1], {n, 0, 30}] // Flatten (* _Michael De Vlieger_, Aug 01 2016 *)
%Y A274660 Cf. A099774, A228443, A274658.
%K A274660 sign,easy,tabf
%O A274660 0,3
%A A274660 _Wolfdieter Lang_, Jul 27 2016