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A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.

%I A109974 #48 Aug 07 2025 08:36:28
%S A109974 1,2,1,2,3,1,3,4,5,1,2,7,10,9,1,4,6,21,28,17,1,2,12,26,73,82,33,1,4,8,
%T A109974 50,126,273,244,65,1,3,15,50,252,626,1057,730,129,1,4,13,85,344,1394,
%U A109974 3126,4161,2188,257,1,2,18,91,585,2402,8052,15626,16513,6562,513,1
%N A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.
%C A109974 Rows sums are A108639. Antidiagonal sums are A109976. Matrix inverse is A109977.
%C A109974 From _Wolfdieter Lang_, Jan 29 2016: (Start)
%C A109974 The sum of the (k-1)th power of the divisors of n, sigma_(k-1)(n), appears also as eigenvalue lambda(k, n) of the Hecke operators T_n, n a positive integer, acting on the normalized Eisenstein series E_k(q) = ((2*Pi*i)^k/((k-1)!*Zeta(k))*G_k(q) with even k >= 4 and q = 2*Pi*i*z, where z is from the upper half of the complex plane: T_n E_k = sigma_(k-1)(n)*E_k. These Eisenstein series are entire modular forms of weight k, and each E_k(q) is a simultaneous eigenform of the Hecke operators T_n, for every n >= 1.
%C A109974 This results from the Fourier coefficients of E_k(q) = Sum_{m>=0} E(k, m)*q^m, with E(k, 0) =1 and E(k, m) = ((2*Pi*i)^k / ((k-1)!*Zeta(k))* sigma_(k-1)(m) for m >= 1, together with the Fourier coefficients of T_n E_k. The eigenvalues lambda(n, k) = (Sum_{d | gcd(n,m)} d^{k-1}*E(k, m*n/d^2)) / E(k, m) for each m >= 0. For m=0 this becomes lambda(n, k) = sigma_(k-1)(n).
%C A109974 For Hecke operators, Fourier coefficients and simultaneous eigenforms see, e.g., the Koecher - Krieg reference, p. 207, eqs. (5) and (6) and p. 211, section 4, or the Apostol reference, p. 120, eq. (13), pp. 129 - 134. (End)
%D A109974 Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 120, 129 - 134.
%D A109974 Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
%D A109974 Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.
%H A109974 Alois P. Heinz, <a href="/A109974/b109974.txt">Antidiagonals k = 0..140, flattened</a>
%F A109974 Regarded as a triangle, T(n, k) = if(k<=n, sigma(k-1, n-k+1), 0). - _Franklin T. Adams-Watters_, Jul 17 2006
%F A109974 If the row index (the index of the antidiagonal of the array) is taken as m with offset 1 the triangle is T(m, k) = sigma_k(m-k), 1 <= k+1 <= m, otherwise 0. - _Wolfdieter Lang_, Jan 14 2016
%F A109974 G.f. for the triangle with offset 1: G(x,y) = Sum_{j>=1} x^j/((1-x^j)*(1-j*x*y)). - _Robert Israel_, Jan 14 2016
%e A109974 Start of array:
%e A109974   1,  2,  2,   3,   2,    4, ...
%e A109974   1,  3,  4,   7,   6,   12, ...
%e A109974   1,  5, 10,  21,  26,   50, ...
%e A109974   1,  9, 28,  73, 126,  252, ...
%e A109974   1, 17, 82, 273, 626, 1394, ...
%e A109974   ...
%e A109974 The triangle T(m, k) with row offset 1 starts:
%e A109974   m\k 0  1  2   3    4    5    6    7   8  9 ...
%e A109974   1:  1
%e A109974   2:  2  1
%e A109974   3:  2  3  1
%e A109974   4:  3  4  5   1
%e A109974   5:  2  7 10   9    1
%e A109974   6:  4  6 21  28   17    1
%e A109974   7:  2 12 26  73   82   33    1
%e A109974   8:  4  8 50 126  273  244   65    1
%e A109974   9:  3 15 50 252  626 1057  730  129   1
%e A109974   10: 4 13 85 344 1394 3126 4161 2188 257  1
%e A109974   ... - _Wolfdieter Lang_, Jan 14 2016
%p A109974 with(numtheory):
%p A109974 seq(seq(sigma[k](1+d-k), k=0..d), d=0..12);  # _Alois P. Heinz_, Feb 06 2013
%t A109974 rows=12; Flatten[Table[DivisorSigma[k-n, n], {k,1,rows}, {n,k,1,-1}]] (* _Jean-François Alcover_, Nov 15 2011 *)
%o A109974 (Magma)
%o A109974 A109974:= func< n,k | DivisorSigma(k-1, n-k+1) >;
%o A109974 [A109974(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 18 2023
%o A109974 (SageMath)
%o A109974 def A109974(n,k): return sigma(n-k+1, k-1)
%o A109974 flatten([[A109974(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 18 2023
%Y A109974 Rows: A000005, A000203, A001157, A001158, A001159, A001160, A013954 - A013972.
%Y A109974 Columns: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660.
%Y A109974 Row sums A108639.
%Y A109974 Diagonals A082245, A023887.
%Y A109974 Related sequences: A082771, A109976, A109977, A109978.
%K A109974 easy,nonn,tabl,nice
%O A109974 0,2
%A A109974 _Paul Barry_, Jul 06 2005