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A096962 Sum_{0<d|n, n/d odd} d^9. 7

%I

%S 1,512,19684,262144,1953126,10078208,40353608,134217728,387440173,

%T 1000000512,2357947692,5160042496,10604499374,20661047296,38445332184,

%U 68719476736,118587876498,198369368576,322687697780,512000262144

%N Sum_{0<d|n, n/d odd} d^9.

%H Seiichi Manyama, <a href="/A096962/b096962.txt">Table of n, a(n) for n = 1..10000</a>

%H H. H. Chan and C. Krattenthaler, <a href="https://arxiv.org/abs/math/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.

%F G.f.: Sum_{k>0} k^9 * x^k / (1 - x^(2*k)).

%e G.f. = q + 512*q^2 + 19684*q^3 + 262144*q^4 + 1953126*q^5 + 10078208*q^6 + ...

%t a[ n_] := If[ n < 1, 0, Sum[ d^9 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* _Michael Somos_, Jun 04 2013 *)

%t a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^4, u4 = QPochhammer[ q^4]^8}, q u2 (u1 + 32 q u4) (u1^2 + 496 q u4 u1 + 7936 q^2 u4^2 ) / u1], {q, 0, n}]; (* _Michael Somos_, Jun 04 2013 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^9))}; /* _Michael Somos_, Jun 04 2013 */

%o (PARI) {a(n) = local(A, A1, A2, A4); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^4; A4 = eta(x^4 + A)^8; polcoeff( A2 * (A1 + 32*x * A4) * (A1^2 + 496*x * A1*A4 + 7936*x^2 * A4^2) / A1, n))}; /* _Michael Somos_, Jun 04 2013 */

%o (Sage) ModularForms( Gamma0(2), 10, prec=33).2; # _Michael Somos_, Jun 04 2013

%o (MAGMA) A := Basis( ModularForms( Gamma0(2), 10), 21); A[2] + 512*A[3]; /* _Michael Somos_, Aug 25 2014 */

%Y Cf. A007331, A096960, A096961, A096963.

%K nonn,mult

%O 1,2

%A _Ralf Stephan_, Jul 18 2004

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Last modified January 28 13:18 EST 2020. Contains 331321 sequences. (Running on oeis4.)