A096963 a(n) = Sum {0<d|n, n/d odd} d^11.

1, 2048, 177148, 4194304, 48828126, 362799104, 1977326744, 8589934592, 31381236757, 100000002048, 285311670612, 743012564992, 1792160394038, 4049565171712, 8649804864648, 17592186044416, 34271896307634

Offset

1,2

Comments

This is the member k=11 of the k-family sigma^#_k(n) := Sum {0<d|n, n/d odd} d^k.

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

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).

K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94, Case 24, Theorem 8.

Index entries for sequences mentioned by Glaisher.

Formula

G.f.: Sum_{n>0} n^11 * x^n / (1 - x^(2*n)).

a(n) = Sum {0<d|n, n/d odd} d^11.

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 2^(11*e) and a(p^e) = (p^(11*e+11)-1)/(p^11-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^12, where c = 1365*zeta(12)/16384 = 691*Pi^12/7664025600 = 0.0833334904... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-11)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

Example

G.f. = q + 2048*q^2 + 177148*q^3 + 4194304*q^4 + 48828126*q^5 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ d^11 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Nov 30 2014 *)
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 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^11))}; /* Michael Somos, Nov 30 2014 */
(Sage) ModularForms( Gamma0(2), 12, prec=18).3; # Michael Somos, Nov 30 2014
(Magma) A := Basis( ModularForms( Gamma0(2), 12), 18); A[2] + 2048*A[3] + 177148*A[4]; /* Michael Somos, Nov 30 2014 */

Crossrefs

Cf. A007331, A002131, A013670, A096960, A096961, A096962.

Sequence in context: A017686 A008455 A079395 * A234740 A134684 A035769

Adjacent sequences: A096960 A096961 A096962 * A096964 A096965 A096966

Keyword

nonn,mult

Author

Ralf Stephan, Jul 18 2004

Status

approved