A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.
1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 2, 0, 12, 8, 1, 0, 0, 4, 8, 24, 10, 1, 0, 0, 8, 6, 32, 40, 12, 1, 0, 0, 0, 24, 24, 80, 60, 14, 1, 0, 0, 0, 24, 48, 90, 160, 84, 16, 1, 0, 2, 4, 0, 96, 112, 252, 280, 112, 18, 1, 0, 0, 4, 12, 64, 240, 312, 574, 448, 144, 20, 1
Offset: 0
Examples
[ 0] 1, 0, 0, 0, 0, 0, 0 0, 0, 0, ... A000007
[ 1] 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, ... A000122
[ 2] 1, 4, 4, 0, 4, 8, 0, 0, 4, 4, ... A004018
[ 3] 1, 6, 12, 8, 6, 24, 24, 0, 12, 30, ... A005875
[ 4] 1, 8, 24, 32, 24, 48, 96, 64, 24, 104, ... A000118
[ 5] 1, 10, 40, 80, 90, 112, 240, 320, 200, 250, ... A000132
[ 6] 1, 12, 60, 160, 252, 312, 544, 960, 1020, 876, ... A000141
[ 7] 1, 14, 84, 280, 574, 840, 1288, 2368, 3444, 3542, ... A008451
[ 8] 1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, ... A000143
[ 9] 1, 18, 144, 672, 2034, 4320, 7392, 12672, 22608, 34802, ... A008452
[10] 1, 20, 180, 960, 3380, 8424, 16320, 28800, 52020, 88660, ... A000144
A005843, v, A130809, v, A319576, v , ... diagonal: A066535
A046092, A319575, A319577, ...
References
- E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954.
- J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006).
Links
- Seiichi Manyama, Descending antidiagonals n = 0..139, flattened
- L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124.
- S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.
- H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
- Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
- S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
- Index entries for sequences related to sums of squares
Programs
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Maple
A319574row := proc(n, len) series(JacobiTheta3(0, x)^n, x, len+1); [seq(coeff(%, x, j), j=0..len-1)] end: seq(print([n], A319574row(n, 10)), n=0..10); # Alternative, uses function PMatrix from A357368. PMatrix(10, n -> A000122(n-1)); # Peter Luschny, Oct 19 2022
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Mathematica
A[n_, k_] := If[n == k == 0, 1, SquaresR[n, k]]; Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2018 *) -
Sage
for n in (0..10): Q = DiagonalQuadraticForm(ZZ, [1]*n) print(Q.theta_series(10).list())
Comments