A076909 Coefficients of 4-point function in dimension 4.

6, 120960, 4136832000, 148146924602880, 5420219848911544320, 200623934537137119778560, 7478994517395643259712737280, 280135301818357004749298146851840, 10528167289356385699173014219946393600, 396658819202496234945300681212382224722560, 14972930462574202465673643937107499992165427200

Offset: 0

Author

N. J. A. Sloane, Nov 28 2002

Keywords

• nonn

Links

• David R. Morrison, Mathematical Aspects of Mirror Symmetry, arXiv:alg-geom/9609021, 1996; in Complex Algebraic Geometry (J. Kollár, ed.), IAS/Park City Math. Series, vol. 3, 1997, pp. 265-340
• H. Movasati, Foliation.lib.
• H. Movasati, Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016-2017. See (1/6)Y_1^2(q) in Section 8.3.

Programs

SINGULAR

// This program has to be compiled in SINGULAR. By changing "int iter" you can
// calculate more coefficients. Note that this program is using a library calling
// "foliation.lib" written by H. Movasati, which is available in the link given in
// LINKS section as Foliation.lib.
LIB "linalg.lib"; LIB "foliation.lib";
ring r=0, (t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,q),dp;
int pm=1;number t10=1/36;number ko=1/216;number c4=ko^2;number t20=-1;number t81=49/18;number a=-6*t20;
poly dis=t_1^6-t_6;
poly dt1=dis*(-t_1*t_2+t_3);
poly dt2=(1296*c4*t_3^2*t_4*t_8-t_1^6*t_2^2+t_2^2*t_6);
poly dt3=(1296*c4*t_3^2*t_5*t_8-3*t_1^6*t_2*t_3+3*t_2*t_3*t_6);
poly dt4=(-1296*c4*t_3^2*t_7*t_8-t_1^6*t_2*t_4+t_2*t_4*t_6);
poly dt5=(1296*c4*t_3*t_5^2*t_8-4*t_1^6*t_2*t_5-2*t_1^6*t_3*t_4+ 5*t_1^4*t_3*t_8+4*t_2*t_5*t_6+2*t_3*t_4*t_6)/(2);
poly dt6=dis*(-6*t_2*t_6);
poly dt7=dis*((1296*c4*t_4^2-t_1^2)/(2592*c4));
poly dt8=(-3*t_1^6*t_2*t_8+3*t_1^5*t_3*t_8+3*t_2*t_6*t_8);
list pose;
pose=(60*ko)/(49*t10^2)*t81*q+(t10),(-162*t20*ko)/(49*t10^3)*t81*q+(t20),(-66*t20*ko)/(7*t10^2)*t81*q+(t10*t20),16/(147*t10^2)*t81*q+(-t10)/(36*ko),45/(49*t10)*t81*q+(-t10^2)/(12*ko),(3888*t10^3*ko)/49*t81*q,1/(1512*t10*t20*ko)*t81*q+(-t10^2)/(1296*t20*ko^2),t81*q+(-t10^3)/(36*ko);
list vecfield=dt1,dt2,dt3,dt4,dt5,dt6,dt7,dt8;
list denomv=dis,dis,dis,dis,dis,dis,dis,dis;
intvec upto=1,1,1,1,1,1,1,1;intvec whichpow;
int iter=10;
int n;
for (n=2; n<=iter;n=n+1){upto=n,n,n,n,n,n,n,n; whichpow=upto;pose=qexpansion(vecfield,denomv,pose,upto,upto,a); n;}
poly y=1/216*pose[3]^4*OneOver(pose[1]^6-pose[6],std(ideal(q^(iter+1))),iter+1);
reduce(y,std(ideal(q^(iter+1))));
/* Younes Nikdelan, Mar 28 2018 */

Formula

G.f.: ((-1/36 + 14*A300196)^4)/(216((1/36 + 20*A300194)^6+1/46656 * A300199)), where the sequence numbers stand for the generating functions of the respective sequences. This is from equation (7.13) of the Movasati & Nikdelan link. - Younes Nikdelan, Mar 28 2018

Extensions

a(8)-a(10) from Younes Nikdelan, Feb 28 2018