login
Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0.
26

%I #45 Oct 18 2023 02:10:52

%S 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 50,126,273,244,65,1,3,15,50,252,626,1057,730,129,1,4,13,85,344,1394,

%U 3126,4161,2188,257,1,2,18,91,585,2402,8052,15626,16513,6562,513,1

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

%C Rows sums are A108639. Antidiagonal sums are A109976. Matrix inverse is A109977.

%C From _Wolfdieter Lang_, Jan 29 2016: (Start)

%C 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 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 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 Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 120, 129 - 134.

%D Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.

%H Alois P. Heinz, <a href="/A109974/b109974.txt">Antidiagonals k = 0..140, flattened</a>

%F 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 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 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 Start of array:

%e 1, 2, 2, 3, 2, 4, ...

%e 1, 3, 4, 7, 6, 12, ...

%e 1, 5, 10, 21, 26, 50, ...

%e 1, 9, 28, 73, 126, 252, ...

%e 1, 17, 82, 273, 626, 1394, ...

%e ...

%e The triangle T(m, k) with row offset 1 starts:

%e m\k 0 1 2 3 4 5 6 7 8 9 ...

%e 1: 1

%e 2: 2 1

%e 3: 2 3 1

%e 4: 3 4 5 1

%e 5: 2 7 10 9 1

%e 6: 4 6 21 28 17 1

%e 7: 2 12 26 73 82 33 1

%e 8: 4 8 50 126 273 244 65 1

%e 9: 3 15 50 252 626 1057 730 129 1

%e 10: 4 13 85 344 1394 3126 4161 2188 257 1

%e ... - _Wolfdieter Lang_, Jan 14 2016

%p with(numtheory):

%p seq(seq(sigma[k](1+d-k), k=0..d), d=0..12); # _Alois P. Heinz_, Feb 06 2013

%t rows=12; Flatten[Table[DivisorSigma[k-n, n], {k,1,rows}, {n,k,1,-1}]] (* _Jean-François Alcover_, Nov 15 2011 *)

%o (Magma)

%o A109974:= func< n,k | DivisorSigma(k-1, n-k+1) >;

%o [A109974(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 18 2023

%o (SageMath)

%o def A109974(n,k): return sigma(n-k+1, k-1)

%o flatten([[A109974(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 18 2023

%Y Rows: A000005, A000203, A001157, A001158, A001159, A001160, A013954 - A013972.

%Y Columns: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660.

%Y Row sums A108639.

%Y Diagonals A082245, A023887.

%Y Related sequences: A082771, A109976, A109977, A109978.

%K easy,nonn,tabl,nice

%O 0,2

%A _Paul Barry_, Jul 06 2005