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A109974 Array read by downwards antidiagonals: sigma_k(n) for n >= 1, k >= 0. 24
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, 50, 126, 273, 244, 65, 1, 3, 15, 50, 252, 626, 1057, 730, 129, 1, 4, 13, 85, 344, 1394, 3126, 4161, 2188, 257, 1, 2, 18, 91, 585, 2402, 8052, 15626, 16513, 6562, 513, 1, 6, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

From Wolfdieter Lang, Jan 29 2016: (Start)

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.

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).

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)

REFERENCES

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

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

LINKS

Alois P. Heinz, Antidiagonals k = 0..140, flattened

FORMULA

Regarded as a triangle, T(n, k) = if(k<=n, sigma(k-1, n-k+1), 0). - Franklin T. Adams-Watters, Jul 17 2006

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

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

EXAMPLE

Start of array:

1,  2,  2,   3,   2,    4, ...

1,  3,  4,   7,   6,   12, ...

1,  5, 10,  21,  26,   50, ...

1,  9, 28,  73, 126,  252, ...

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

...

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

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

1:  1

2:  2  1

3:  2  3  1

4:  3  4  5   1

5:  2  7 10   9    1

6:  4  6 21  28   17    1

7:  2 12 26  73   82   33    1

8:  4  8 50 126  273  244   65    1

9:  3 15 50 252  626 1057  730  129   1

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

... - Wolfdieter Lang, Jan 14 2016

MAPLE

with(numtheory):

seq(seq(sigma[k](1+d-k), k=0..d), d=0..12);  # Alois P. Heinz, Feb 06 2013

MATHEMATICA

rows = 12; Flatten[ Table[ DivisorSigma[k-n, n], {k, 1, rows}, {n, k, 1, -1}]] (* Jean-Fran├žois Alcover, Nov 15 2011 *)

CROSSREFS

Rows: A000005, A000203, A001157, A001158, A001159, A001160, A013954-A013972; columns: A000051, A034472, A001576, A034474, A034488, A034491, A034496, A034513, A034517, A034524, A034660; row sums A108639; diagonals A082245, A023887; also see A082771, A109976, A109977, A109978.

Sequence in context: A036838 A066010 A209556 * A213008 A215520 A026820

Adjacent sequences:  A109971 A109972 A109973 * A109975 A109976 A109977

KEYWORD

easy,nonn,tabl,nice

AUTHOR

Paul Barry, Jul 06 2005

STATUS

approved

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Last modified October 16 20:23 EDT 2019. Contains 328103 sequences. (Running on oeis4.)