cp's OEIS Frontend

A307618 A Calabi-Yau period integral: a(n) = C(4*n,2*n)*C(2*n,n)^3.

%I A307618 #30 Jan 25 2023 11:47:12
%S A307618 1,48,15120,7392000,4414410000,2956651746048,2133278987583744,
%T A307618 1621682968820428800,1281351259836532170000,1043032815185819858400000,
%U A307618 869343653096068540955685120,738637974389826550020188712960,637665137404661719206664998969600
%N A307618 A Calabi-Yau period integral: a(n) = C(4*n,2*n)*C(2*n,n)^3.
%C A307618 Entry number six in the "Big Table" of Almkvist et al. (see links). The period T(x) = Sum_{n>=0} a(n)*x^(2*n) is also the first x-derivative of the 6-volume associated to the algebraic variety V6 = P1 & P2 & P3, with P1 : X1^2 + Y1^2 = X2^2 + Y2^2, P2 : X2^2 + Y2^2 = X3^2 + Y3^2, P3 : x=(X1^2 + X2^2 + X3^2 + Y1^2 + Y2^2 + Y3^2)^3*(1 - X1*X2*X3*Y1*Y2*Y3). The small x limit reduces V6 to a 6-ball with 6-volume proportional to x. Similar constructions are known to exist for a few other geometries on Almkvist's list, most notably #3: A186420, and #16: A039699.
%H A307618 G. Almkvist et al., <a href="https://arxiv.org/abs/math/0507430">Tables of Calabi-Yau Equations</a>, arXiv:math/0507430 [math.AG], 2005-2010, p. 10.
%H A307618 G. Almkvist et al., <a href="https://www.staff.uni-mainz.de/vanstrat/Zoek/sequences.txt">Raw Calabi-Yau Period Data</a>, Johannes Gutenberg Universität Mainz, 2019, case 1.6.
%F A307618 G.f.: 4F3({1/4, 3/4, 1/2, 1/2}, {1, 1, 1}, 1024*x).
%F A307618 Define the period integral:
%F A307618 dt(x) = dz1*dz2*dz3/sqrt(1-32*x*cos(z1)*cos(z2)*cos(z3)).
%F A307618 T(x)=1/(2*Pi)^3*Integral_{0..2*Pi,0..2*Pi,0..2*Pi} dt(x),
%F A307618 the Picard-Fuchs coefficients:(c0,c1,c2,c3,c4)=
%F A307618 (768*x, 14592*x^2-1, x*(25344*x^2-7), 2*x^2*(5120*x^2-3), x^3*(32*x-1)*(32*x+1)),
%F A307618 and the certificate function:
%F A307618 G(z1,z2,z3)=(16*sin(z1)*(
%F A307618        48*x*cos(z1)
%F A307618      + cos(z2)*cos(z3)
%F A307618      + 48*x*cos(z1)*(cos(z3)^2 + cos(z2)^2)
%F A307618      + 2304*x^2*cos(z1)^2*cos(z2)*cos(z3)
%F A307618      + 80*x*cos(z1)*cos(z2)^2*cos(z3)^2
%F A307618      + 384*x^2*cos(z1)^2*(cos(z2)*cos(z3)^3 + cos(z2)^3*cos(z3))
%F A307618      + 256*x^2*cos(z1)^2*cos(z2)^3*cos(z3)^3)
%F A307618     )/(3*(1 - 32*x*cos(z1)*cos(z2)*cos(z3))^(7/2)),
%F A307618 Then: 0 = Sum_{n=0..4}cn*d^n/dx^n dt(x) + d/dz1 G(z1,z2,z3) + d/dz2 G(z2,z3,z1) + d/dz3 G(z3,z1,z2), thus: 0 = Sum_{n=0..4} cn*d^n/dx^n T(x).
%F A307618 Furthermore, let (a1,a2,a3)=(c1,c2,c3)/c0, then also: 0 = (1/2)*a2*a3 - (1/8)*a3^3 + d/dx(a2) - (3/4)*a3*d/dx(a3) - (1/2)*d^2/dx^2(a3) - a1.
%F A307618 D-finite with recurrence: n^4*a(n) -16*(4*n-1)*(4*n-3)*(-1+2*n)^2*a(n-1)=0. - _R. J. Mathar_, Jan 27 2020
%t A307618 Binomial[4*#,2*#]*Binomial[2*#,#]^3&/@Range[0,10]
%Y A307618 Hadamard Factors: A000984, A002894, A002897, A001448, A000897, A008977.
%Y A307618 Calabi-Yau Periods: A008978, A186420, A268553, A039699.
%K A307618 nonn
%O A307618 0,2
%A A307618 _Bradley Klee_, Jun 04 2019