A005927 - OEIS

%I M3262 #21 Mar 08 2020 08:40:08

%S 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,

%T 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,

%U 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

%N Theta series of diamond with respect to deep hole.

%C 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).

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

%H G. C. Greubel, <a href="/A005927/b005927.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H N. J. A. Sloane and B. K. Teo, <a href="http://dx.doi.org/10.1063/1.449551">Theta series and magic numbers for close-packed spherical clusters</a>, J. Chem. Phys. 83 (1985) 6520-6534.

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

%F 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

%F 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

%F 4 * A008443(n) = a(8*n + 3). A005887(n) = a(8*n + 4). - _Michael Somos_, Aug 17 2009

%e 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

%t 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 *)

%o (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 */

%K nonn

%O 0,4

%A _N. J. A. Sloane_