A298931 -id:A298931 - cp's OEIS frontend

A005872 Theta series of hexagonal close-packing with respect to octahedral hole.

0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 18, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 24, 0, 6, 0, 0, 0, 36, 0, 12, 0, 0, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 0, 0, 18

Offset: 0

N. J. A. Sloane

			G.f. = 6*q^3 + 6*q^9 + 6*q^11 + 12*q^15 + 6*q^17 + 12*q^23 + 18*q^27 + ... - _Michael Somos_, Jul 06 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

Sean A. Irvine, Table of n, a(n) for n = 0..999

Cf. A298931.

a[ n_] := SeriesCoefficient[ 6 x^3 QPochhammer[ x^16]^2 QPochhammer[ x^18]^3 / (QPochhammer[ x^6] QPochhammer[ x^8]), {x, 0, n}]; (* Michael Somos, Jul 06 2018 *)

{a(n) = my(A, m); if( n<3 || n%2==0, 0, m = n\2 - 1; A = x * O(x^m); 6 * polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), m))}; /* Michael Somos, Jul 06 2018 */

a(2*n) = 0. a(2*n + 3) = 6*A298931(n). - Michael Somos, Jul 06 2018

A298932 Expansion of f(-x^3)^3 * phi(-x^12) / (f(-x) * chi(-x^4)) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 4, 0, 4, 4, 6, 0, 5, 3, 6, 0, 6, 4, 4, 0, 8, 4, 6, 0, 9, 6, 6, 0, 6, 6, 12, 0, 8, 4, 12, 0, 8, 7, 8, 0, 9, 6, 8, 0, 12, 8, 6, 0, 8, 6, 14, 0, 12, 6, 12, 0, 8, 8, 12, 0, 13, 6, 12, 0, 18, 10, 8, 0, 8, 12, 12, 0, 16, 7, 14, 0, 12, 8, 12, 0, 16

Offset: 0

Michael Somos, Jan 29 2018

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^8 + 4*x^9 + ...
G.f. = q + q^3 + 2*q^5 + 3*q^9 + 2*q^11 + 4*q^13 + 4*q^17 + 4*q^19 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A213607, A298931, A298933.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 QPochhammer[ -x^4, x^4] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Expansion of q^(-1/2) * eta(q^3)^3 * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [1, 1, -2, 2, 1, -2, 1, 1, -2, 1, 1, -3, 1, 1, -2, 1, 1, -2, 1, 2, -2, 1, 1, -3, ...].
a(4*n + 3) = 0. a(3*n + 2) = 2 * A213607(n). a(n) = A298931(3*n). a(2*n) = A298933(n).

cp's OEIS Frontend

A005872 Theta series of hexagonal close-packing with respect to octahedral hole.

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