A005927 Theta series of diamond with respect to deep hole. (Formerly M3262)

0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 12, 8, 0, 0, 0, 0, 0, 0, 12, 24, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 24, 30, 0, 0, 0, 0, 0, 0, 12, 24, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 12, 48, 0, 0, 0, 0, 0, 0, 28, 24, 0, 0, 0, 0, 0, 0, 36, 48, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0

Offset

0,4

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

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.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of 4 * q^3 * psi^3(q^8) + (phi^3(q^4) - phi^3(-q^4)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 17 2009

a(8*n + 0) = a(8*n + 1) = a(8*n + 2) = a(8*n + 5) = a(8*n + 6) = a(8*n + 7) = 0. - Michael Somos, Aug 17 2009

4 * A008443(n) = a(8*n + 3). A005887(n) = a(8*n + 4). - Michael Somos, Aug 17 2009

Example

4*q^3 + 6*q^4 + 12*q^11 + 8*q^12 + 12*q^19 + 24*q^20 + 16*q^27 + ... - Michael Somos, Aug 17 2009

Mathematica

a[n_]:= SeriesCoefficient[4*q^3*QPochhammer[-q^8, q^8]^3* QPochhammer[q^16, q^16]^3 + (EllipticTheta[3, 0, q^4]^3 - EllipticTheta[3, 0, -q^4]^3)/2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 01 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, if( n%8 == 3, n \= 8; polcoeff( 4 * sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n), if( n%8 == 4, n /= 4; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n))^3, n), 0 )))} /* Michael Somos, Aug 17 2009 */

Crossrefs

Sequence in context: A299639 A244444 A231407 * A201529 A079207 A259825

Adjacent sequences: A005924 A005925 A005926 * A005928 A005929 A005930

Keyword

nonn

Author

N. J. A. Sloane

Status

approved