A096961
a(n) = Sum_{0
1, 128, 2188, 16384, 78126, 280064, 823544, 2097152, 4785157, 10000128, 19487172, 35848192, 62748518, 105413632, 170939688, 268435456, 410338674, 612500096, 893871740, 1280016384, 1801914272, 2494358016, 3404825448
Offset: 1
Examples
G.f. = q + 128*q^2 + 2188*q^3 + 16384*q^4 + 78126*q^5 + 280064*q^6 + 823544*q^7 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- 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.
- J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
- Index entries for sequences mentioned by Glaisher.
Programs
-
Magma
A := Basis( ModularForms( Gamma0(2), 8), 24); A[2] + 128*A[3]; /* Michael Somos, Nov 30 2014 */
-
Mathematica
a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^8, u4 = QPochhammer[ q^4]^8}, q u2 (u1^2 + 136 q u4 u1 + 2176 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *) a[ n_] := If[ n < 1, 0, Sum[ d^7 Mod[ n/d, 2], {d, Divisors[ n]}]]; (* Michael Somos, Jan 09 2015 *) -
PARI
{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^7))}; -
Sage
ModularForms( Gamma0(2), 8, prec=24).2; # Michael Somos, Jun 04 2013
Formula
G.f.: Sum_{k>0} k^7 * x^k / (1 - x^(2*k)).
Expansion of (E_8(q) - E_8(q^2)) / 480 in powers of q where E_8() is an Eisenstein series (A008410). - Michael Somos, Jan 09 2015
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(7*e) and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 255*zeta(8)/2048 = 17*Pi^8/1290240 = 0.125019... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-7)*(1-1/2^s). - Amiram Eldar, Jan 09 2023