The On-Line Encyclopedia of Integer Sequences (OEIS)

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A302860

a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.
(history; published version)

#9 by Vaclav Kotesovec at Sun Apr 15 02:55:55 EDT 2018

STATUS

reviewed

approved

#8 by Joerg Arndt at Sun Apr 15 02:13:03 EDT 2018

STATUS

proposed

reviewed

#7 by Vaclav Kotesovec at Sat Apr 14 17:11:37 EDT 2018

STATUS

editing

proposed

#6 by Vaclav Kotesovec at Sat Apr 14 17:11:15 EDT 2018

FORMULA

a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018

#5 by Vaclav Kotesovec at Sat Apr 14 17:00:02 EDT 2018

CROSSREFS

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

STATUS

proposed

editing

#4 by Ilya Gutkovskiy at Sat Apr 14 16:03:10 EDT 2018

STATUS

editing

proposed

#3 by Ilya Gutkovskiy at Sat Apr 14 15:10:00 EDT 2018

CROSSREFS

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861.

#2 by Ilya Gutkovskiy at Sat Apr 14 15:03:41 EDT 2018

NAME

allocated for Ilya Gutkovskiya(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

DATA

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457

OFFSET

0,2

COMMENTS

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).

LINKS

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

<a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

FORMULA

a(n) = A122510(n,n).

MATHEMATICA

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]

Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

CROSSREFS

Main diagonal of A122510.

Cf. A000122, A001650, A046895, A057655, A066535, A117609.

KEYWORD

allocated

nonn

AUTHOR

Ilya Gutkovskiy, Apr 14 2018

STATUS

approved

editing

#1 by Ilya Gutkovskiy at Sat Apr 14 15:03:41 EDT 2018

NAME

allocated for Ilya Gutkovskiy

KEYWORD

allocated

STATUS

approved