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A096963 a(n) = Sum {0

%I A096963 #27 Jan 09 2023 01:49:59
%S A096963 1,2048,177148,4194304,48828126,362799104,1977326744,8589934592,
%T A096963 31381236757,100000002048,285311670612,743012564992,1792160394038,
%U A096963 4049565171712,8649804864648,17592186044416,34271896307634
%N A096963 a(n) = Sum {0<d|n, n/d odd} d^11.
%C A096963 This is the member k=11 of the k-family sigma^#_k(n) := Sum {0<d|n, n/d odd} d^k.
%C A096963 This notation appears in the Ono et al. link, Theorem 5 (with k=3, see A007331) and Theorem 8 (with k=11). - _Wolfdieter Lang_, Jan 13 2017
%H A096963 Seiichi Manyama, <a href="/A096963/b096963.txt">Table of n, a(n) for n = 1..10000</a>
%H A096963 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.
%H A096963 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A096963 K. Ono, S. Robins and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94, Case 24, Theorem 8.
%H A096963 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A096963 G.f.: Sum_{n>0} n^11 * x^n / (1 - x^(2*n)).
%F A096963 a(n) = Sum {0<d|n, n/d odd} d^11.
%F A096963 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A096963 Multiplicative with a(2^e) = 2^(11*e) and a(p^e) = (p^(11*e+11)-1)/(p^11-1) for p > 2.
%F A096963 Sum_{k=1..n} a(k) ~ c * n^12, where c = 1365*zeta(12)/16384 = 691*Pi^12/7664025600 = 0.0833334904... . (End)
%F A096963 Dirichlet g.f.: zeta(s)*zeta(s-11)*(1-1/2^s). - _Amiram Eldar_, Jan 09 2023
%e A096963 G.f. = q + 2048*q^2 + 177148*q^3 + 4194304*q^4 + 48828126*q^5 + ...
%t A096963 a[ n_] := If[ n < 1, 0, Sum[ d^11 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* _Michael Somos_, Nov 30 2014 *)
%t A096963 a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u4 = QPochhammer[ q^4]^8}, q (u1^4 + 2072 q u4 u1^3 + 210048 q^2 u4^2 u1^2 + 5660672 q^3 u4^3 u1 + 45285376 q^4 u4^4) / u1 ], {q, 0, n}]; (* _Michael Somos_, Nov 30 2014 *)
%o A096963 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^11))}; /* _Michael Somos_, Nov 30 2014 */
%o A096963 (Sage) ModularForms( Gamma0(2), 12, prec=18).3; # _Michael Somos_, Nov 30 2014
%o A096963 (Magma) A := Basis( ModularForms( Gamma0(2), 12), 18); A[2] + 2048*A[3] + 177148*A[4]; /* _Michael Somos_, Nov 30 2014 */
%Y A096963 Cf. A007331, A002131, A013670, A096960, A096961, A096962.
%K A096963 nonn,mult
%O A096963 1,2
%A A096963 _Ralf Stephan_, Jul 18 2004