A076909 - OEIS

A076909

Coefficients of 4-point function in dimension 4.

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

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OFFSET

0,1

LINKS

Table of n, a(n) for n=0..10.

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.

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

PROG

(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 */

CROSSREFS

Sequence in context: A003834 A152296 A358811 * A172864 A352022 A262187

Adjacent sequences: A076906 A076907 A076908 * A076910 A076911 A076912