

A000156


Number of ways of writing n as a sum of 24 squares.


3



1, 48, 1104, 16192, 170064, 1362336, 8662720, 44981376, 195082320, 721175536, 2319457632, 6631997376, 17231109824, 41469483552, 93703589760, 200343312768, 407488018512, 793229226336, 1487286966928, 2697825744960, 4744779429216
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OFFSET

0,2


COMMENTS

The Carlitz paper has the wrong formula on p. 505, eq. (3). The factor in front of tau(n/2) should be 2^16 (not 2^12). The mistake appeared in the previous equation derived from eq. (2) where theta_3^(24) * 256*k^4*k'^4 was replaced by 2^8*g(q^2) which produces the factor 2^8*256 = 2^16. (One should subtract on p. 504 the second equation in the middle from the negative of the first equation. There is also a sign mistake in the sum term of the third equation from the bottom.)  Wolfdieter Lang, Sep 24 2016


REFERENCES

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 195, eq. (15.1).
E. Grosswald, Representations of Integers as Sums of Squares. SpringerVerlag, NY, 1985, p. 107.
G. H. Hardy, Ramanujan, 1940, Cambridge, reprinted with additional corrections and comments by AMS Chelsea Publishing, 1999, 2002, Providence, Rhode Island, ch. IX., pp. 153155.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.


LINKS

ShiChao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 20282032.


FORMULA

For n >= 1: a(n) = (16*sigma^*_{11}  128*(512*tau(n/2) + (1)^n*259*tau(n)))/691, with sigma^*_{11} = sigma_{11}^{e}(n)  sigma_{11}^{o}(n) if n even and sigma_{11}(n) otherwise. Here sigma_{11}(n) = A013959(n) and 0 if n is not an integer, sigma_{11}^{e}(n) and sigma_{11}^{o}(n) are the sums of the 11th power of the odd and even positive divisors of n, respectively. Ramanujan's tau(n) = A000594(n) and 0 if n is not an integer. This is from Hardy, ch. IX., p. 155, eqs. (9.17.1) and (9.17.2), and p.142 for the definition of sigma^*_{nu}(n). See also the Ash and Gross reference.
Another version, see the corrected Carlitz reference:
a(n) = (2^4*(sigma_{11}(n) 2*sigma_{11}(n/2) + 2^{12}*sigma_{11}(n/4))  2^7*259*(1)^n*tau(n)  2^16*tau(n/2))/691, n >= 1.
(End)


MAPLE

(sum(x^(m^2), m=10..10))^24; seq(coeff(%, x, n), n=0..30);
# Alternative:
A000156list := proc(len) series(JacobiTheta3(0, x)^24, x, len+1);
seq(coeff(%, x, j), j=0..len1) end: A000156list(21); # Peter Luschny, Oct 02 2018


MATHEMATICA

Table[SquaresR[24, n], {n, 0, 20}] (* Ray Chandler, Nov 28 2006 *)


PROG

(PARI) first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1, sqrtint(n), x^k^2) + 1)^24) \\ Charles R Greathouse IV, Jul 29 2016


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



