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A274658 Irregular triangle which lists in row n the divisors of 2*n+1.

%I A274658 #26 May 02 2025 02:53:46
%S A274658 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,
%T A274658 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,
%U A274658 3,5,9,15,45
%N A274658 Irregular triangle which lists in row n the divisors of 2*n+1.
%C A274658 The length of row n is A099774(n+1).
%C A274658 This gives the odd numbered rows of the irregular triangle A027750.
%C A274658 The row sums are given in A008438.
%C A274658 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.
%H A274658 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 A274658 T(n, k) = k-th divisor of 2*n+1 in increasing order.
%e A274658 The irregular triangle T(n, k) begins:
%e A274658   n, 2n+1\k 1  2   3   4 ...
%e A274658   0,   1:   1
%e A274658   1,   3:   1  3
%e A274658   2,   5:   1  5
%e A274658   3,   7:   1  7
%e A274658   4,   9:   1  3   9
%e A274658   5,  11:   1 11
%e A274658   6,  13:   1 13
%e A274658   7,  15:   1  3   5  15
%e A274658   8,  17:   1 17
%e A274658   9,  19:   1 19
%e A274658   10, 21:   1  3   7  21
%e A274658   11, 23:   1 23
%e A274658   12, 25:   1  5  25
%e A274658   13, 27:   1  3   9  27
%e A274658   14, 29:   1 29
%e A274658   15, 31:   1 31
%e A274658   16, 33:   1  3  11  33
%e A274658   17, 35:   1  5   7  35
%e A274658   18, 37:   1 37
%e A274658   19, 39:   1  3  13  39
%e A274658   20, 41:   1 41
%e A274658 ...
%e A274658 The above mentioned second factor in the sn formula has as q^4 coefficient: sin(1*v) + sin(3*v) + sin(9*v).
%t A274658 Table[Divisors[2 n + 1], {n, 0, 22}] // Flatten (* _Michael De Vlieger_, Jul 18 2016 *)
%o A274658 (PARI) row(n) = divisors(2*n+1); \\ _Amiram Eldar_, May 02 2025
%Y A274658 Cf. A008438 (row sums), A027750, A099774 (row lengths), A274659.
%K A274658 nonn,tabf,easy
%O A274658 0,3
%A A274658 _Wolfdieter Lang_, Jul 18 2016